





































































































































































































































































COPYRIGHT DEPOSIT. 















MATHEMATICAL THEORY 

OF FINANCE 


BY 

T. M. PUTNAM, Ph.D. 

M 

Professor of Mathematics, University of California 


NEW YORK 

JOHN WILEY & SONS, Inc. 

London: Chapman & Hall, Limited 

1923 



H Fit 



Copyright, 1923, by 

T. M. PUTNAM 



M 23 *23 


PRESS OF 

BRAUNWORTH & CO. 
BOOK MANUFACTURERS 
BROOKLYN, N. Y. 


©C1A704068 

/ 




PREFACE 


This book has been prepared primarily to meet the needs 
of students in schools and colleges of Commerce and Business 
Administration. It will also meet the needs of anyone who 
desires a knowledge of the mathematical treatment of financial 
problems arising in ordinary business procedure. 

The scope and method of the book have been designed for a 
three-hour course for one semester, such as is prescribed in the 
College of Commerce in the University of California. It is 
assumed that the student has had a substantial course in Algebra 
equivalent to two years’ study in the high school and a thor¬ 
ough knowledge of logarithmic computation. 

The author has aimed throughout to emphasize fundamental 
principles and to illustrate them with numerous simple examples. 
Experience in teaching the subject has clearly demonstrated 
that, in the short time that can usually be allotted to the 
course, the student can hardly hope to obtain much more than 
an understanding of basic principles. The more technical 
phases of theory and application should, the author feels, be 
left to later study. 

The author wishes to acknowledge his indebtedness to his 
colleagues, Dr. A. R. Williams and Dr. C. D. Shane, for val¬ 
uable suggestions and criticisms, and in particular to Professor 
B. A. Bernstein, who has greatly aided both in the preparation 
of the manuscript and in the correction of the proof. 

^ T. M. Putnam. 


University of California, 
April, 1923. 


in 







TABLE OF CONTENTS 


CHAPTER I 
Interest 

ARTICLE 

1. Definition of interest. 

2. The rate of interest. 

3. Simple interest. 

4. Ordinary and exact simple interest. 

5. Compound interest. 

6. Formulas of compound interest. 

7. Geometric comparison of simple and of compound interest 

8. Nominal and effective rates of interest. 

9. Continuous compound interest. Force of interest. 

10. Present value. 

11. Discount. 

12. Principle of Equivalence. 


PAGE 

1 

1 

1 

2 

«> 

O 

3 

6 

7 

8 

10 

12 

13 


CHAPTER II 
Annuities 


13. Definition. 16 

14. Notation. 16 

15. Present value of an annuity. 16 

16. Formula for a~^. 19 

17. Formulas for and . 21 

18. Deferred annuities. 23 

19. Annuities due. 23 

20. The annuity that 1 will purchase..'. 25 

21. The annuity that will amount to 1. 26 

22. Perpetuities and capitalization. 28 


CHAPTER III 

Amortization—Sinking Funds 

23. Amortization. 

24. Amortization schedules. 

25. Amount of unpaid principal. 


v 



























vi TABLE OF CONTENTS 

ARTICLE PAGE 

26. Sinking funds. 35 

27. Amount in the sinking fund at any time. 37 

28. Amortization of bonded debt. 38 

29. Depreciation. 39 

30. Other methods of estimating depreciation. 42 

31. Composite life. 44 

32. Valuation of mining property. 45 

CHAPTER IV 
Bonds 

33. Description. 49 

34. The investment rate. 49 

35. Bond formula. 50 

36. Premium and discount. 52 

37. Amortization of the premium. 54 

38. Accumulation of discount. 56 

39. Bonds purchased between interest payment dates. 57 

40. Calculation of the investment rate when the purchase price is 

given. 58 

CHAPTER V 
Probability 

41. Definition of probability. 62 

42. Theorems on arrangement and combination. 64 

43. Mutually exclusive events. 66 

44. Compound events. 67 

45. Probability with repeated trials. 68 

46. Mathematical expectation. 70 

47. Mortality tables. 70 

48. Joint life probabilities. 71 

CHAPTER VI 
Life Annuities 

49. Definition of life annuities. 74 

50. Pure endowment. 74 

51. Computation of life annuity. 75 

52. Commutation columns. 77 





























TABLE OF CONTENTS vii 

ARTICLE PAGE 

53. Deferred annuities. 77 

54. Temporary annuities. 78 

55. Annuities due. 79 

CHAPTER VII 

Elementary Principles of Life Insurance 

56. Introduction. 81 

57. Net single premium. Whole-life policy. 82 

58. Commutation symbols . 83 

59. Annual premiums. 84 

60. Net single premium for term insurance. 85 

61. Net annual premium for term insurance. 86 

62. Endowment insurance . 87 

63. Valuation of policies; reserves. 88 

64. Gross premiums; loading. 89 

65. Conclusion. 90 


v 



















TABLES 


TABLE 

I. The number of each day of the year. 

II. Amount of 1. s(l+i) w . 

III. Present value of 1. v n = (1 +i) ~ n . . . 

IV. Present value of 1 per annum, a = 


V. Amount of 1 per annum. 

VI. Annuity which 1 will buy. 


s n[ — 

l 

1 1 

— — —. . . 

Q n I n I 


1 

VII. Amount of 1 for parts of a year. $ = (1-H) P 

i_ 

VIII. Values of j( P )=p[(l-\-i) p — l]. 

IX. Values of -—. 

J(p) 


X. American Experience Table of Mortality, 

XI. Commutation Columns. 


PAGE 

91 

92 
96 

100 

104 

105 

112 

112 

112 

113 

114 


IX 


















MATHEMATICAL THEORY OF FINANCE 


CHAPTER I 

INTEREST 

1. Definition of interest. —Interest may be defined as money 
paid for the use of borrowed capital, or, as that which is earned 
by the productive investment of capital. In a given trans¬ 
action, the capital involved is referred to as the principal. 

2. The rate of interest. —The rate of interest is the sum 
earned on a unit of principal in a unit of time, the latter, unless 
otherwise specified, being taken as one year. It is customary, 
however, to express the rate as the earning per centum, that is, 
the earning on 100 units of principal. Thus, if each dollar of 
capital earns 6 cents a year, the rate is 6 per centum, meaning 
that $100 would earn $6 per year. Throughout the remainder 
of this book, the abbreviated form, “per cent,” will be used 
instead of the Latin per centum. 

3. Simple interest. —Interest usually becomes due at stated 
intervals and, being of the same nature as capital, may be 
reinvested as additional principal. If, however, one is con¬ 
cerned only with the amount earned by the original capital in 
a given time, and not with the productive reinvestment of 
earnings, the investment is said to bear simple interest. 

With a given principal and fixed rate, simple interest is 
proportional to the time. Thus, if P denote the principal, 
i the rate, and n the time in years, then the interest I is given 
by the formula 


/ = Pni. 


(1) 



2 


INTEREST 


The amount , S, is the sum of the principal and interest, 
hence 

S = P+7 =P(l+m).(2) 

4. Ordinary and exact simple interest. —In calculating 

simple interest for fractional parts of a year, it is frequently 
the practice to base the computation on 360 days to a year. 
When this is done, interest is called ordinary simple interest. 
If the calculations are based on 365 days in a year, it is called 
exact simple interest. 

The exact number of days between two given dates may be 
calculated directly, or found from a table such as Table I, 
for the computation of either exact or ordinary simple interest. 

Example 1.—Find the ordinary and the exact interest on $1500 for 
80 days at 6 per cent. 

The interest for one year is $90. Hence the ordinary simple interest is 

$90X3 8 6°o=S20. 

The exact simple interest is 

$90X3 8 e°5=$19.73. 

Example 2.—Find the time that has elapsed between March 20, 1921, 
and October 17, 1921. 

From Table I it is found that October 17 is the 290th day of the 
year and March 20 is the 79t.h day of the year; hence there are 211 
days between these dates. 

Unless otherwise stated, it will be understood that ordinary simple 
interest is required. 


PROBLEMS 

1. What is the monthly simple interest on a note for $2500 bearing 

5 per cent? Arts. $10.42. 

2. Find the time a note for $1850, bearing 6 per cent simple interest, 
would have to run in order to amount to $2000. Ans. 1 yr., 4 mo., 6 days. 

3. Find the ordinary simple interest on $1250 at 7 per cent, from 

April 10 to August 25 of the same year. Ans. $33.30. 

4. Find the exact simple interest in Problem 3. Ans. $32.84. 



COMPOUND INTEREST 


3 


5. Find the ratio of exact interest to ordinary interest, showing that 

it is constant for any number of days. Ans. yf 

6. What is the rate of interest, if $1800 earns $45 in 4 months? 

7. What principal will amount to $1263 in 10 months, 15 days, at 6 per 
cent? 

8. Find the exact simple interest on a note for $1500 bearing 6 per 
cent interest, dated February 4, 1920 (a leap year) and falling due No¬ 
vember 20, 1920. Find also the ordinary simple interest. 

5. Compound interest.—It was seen that simple interest is 
calculated on the original principal only, and is merely pro¬ 
portional to the time. If interest, when due, is added to the 
principal, and interest for the next period is calculated on the 
principal thus increased, this process being continued with each 
succeeding accumulation of interest, then the interest is said 
to be compound. 

It should be observed that, in transactions involving simple 
interest paid at regular intervals, the creditor, collecting his 
interest and investing it at the same rate as in the original loan, 
will accumulate new capital just as rapidly as if he had loaned 
at compound interest originally. 

Because interest is itself of the nature of capital, it becomes 
necessary, in all questions involving equivalence of value, to 
regard all sums as bearing compound interest. This is particu¬ 
larly important when an indebtedness extends over a con¬ 
siderable length of time. 

6. Formulas of compound interest.—Let P be the principal, 
and i the rate of interest. The amount to which the principal 
will accumulate will be denoted by S. The interest for one year 
will be Pi, and the amount at the end of that time will be 
P-\-Pi. This becomes the principal for the second year. 
The interest on it is (P-\-Pi)i, and the amount at the end of the 
second year will be the principal plus the interest earned, or 

P+Pi+(P+Pi)i=P(l+i) 2 . 

Thus, each unit of principal at the beginning of any year will 
accumulate to 1-f-t units at the end of the year, so that the 


4 


INTEREST 


amount may be obtained by multiplying the principal at the 
beginning of the year by 1-f-f. Since P(l+f) 2 is the principal 
at the beginning of the third year, the amount at the end of 
the third year will be P(l+z) 3 . In general, the amount, S, 
at the end of n years, is given by the formula 

£ = P(l+t) n .(3) 


In this process, interest is said to be compounded, or converted, 
annually. It may, however, be compounded semiannually, or 
quarterly, or, in general, m times a year. The time between 
two successive conversions of interest into principal is spoken 
of as the conversion period. 

The same principles as used above apply when the period 
is a fraction of a year, provided merely that i is replaced by the 
interest on one unit of principal for the period in question, 
and n is replaced by the total number of conversion periods. 
Thus, if the rate of interest is 2 per cent per half year, the 
amount of $100 for 10 years is $100(1.02) 20 . It is customary 
to speak of 2 per cent per half year as “ 4 per cent converted 
semiannually,” so that in general if interest is at rate j con¬ 
verted m times a year, Formula (3) is replaced by 


S=P 




In deriving Formulas (3) and (4), the time is supposed to 
contain an integral number of conversion periods. When this 
is not so, it is customary in practice to compute the amount 
at the end of the last conversion period, and then to compute 
simple interest for the fraction of a period remaining. For 
theoretical purposes, however, it is convenient to regard For¬ 
mulas (3) and (4) as true for all values of n, whether integers 
or not. 


Example. —Find the amount at compound interest on $1000, at 4 
per cent converted semiannually, for 2 years and 8 months. Formula 
(4) becomes 

S = $1000(1.02) 16 /3 
= $1111.39 




GEOMETRICAL COMPARISON 


5 


The number of conversion periods is 5. Had the amount been computed 
at the end of the fifth period, it would have been 

$1000(1.02) 5 = $1104.08. 

If simple interest be computed on this sum for the remaining 2 months, it 
is seen to be $7.36. The final amount thus computed is then $1111.44, 
which is slightly in excess of that given by the first method. 

Unless otherwise stated, it will be understood that the amount, S, is 
to be computed by Formulas (3) or (4), whether or not the time contains 
an integral number of periods. 

Table II gives the values of (1-H) n for the usual rates of interest that 
arise in applying either Formula (3) or Formula (4). Thus, for example, 
if the rate of interest is 5 per cent compounded quarterly, there will be 

j 

4 n periods, while —, the rate of interest for the quarterly period, will be 

m 

1* per cent. If n were 5 years, the amount of 1 would be found in the 
column headed If per cent, and opposite the number 20 in the left-hand 
column. The problem is identical with finding the amount of 1 for 20 
years at compound interest at 1 j per cent per year. 

For rates of interest not listed in the table, and for values of n that are 
not integers, the computations may be made by use of logarithms. 

7. Geometrical comparison of simple and compound interest. 

—If the amount, S, be plotted on an ordinary coordinate sys¬ 
tem, time being laid off on the horizontal scale and the corre¬ 
sponding values of S vertically, the simple interest formula 

S =P(l+m) 

is a straight line (Fig. 1). The compound interest amount, 
$ = P(l+f) n , gives a curve which is concave upward, inter¬ 
secting the simple interest line at n = 0 and n= 1. For times 
less than a year, the amount by compound interest is less than 
that by simple interest, but after one year the compound 
amount is larger, on account of the addition of interest to 
principal. • ^ 

When interest is compounded m times a year, the curve 
corresponding to 

5=p(l+i') 

\ m] 


mn 


6 


INTEREST 


bears a corresponding relation to the simple interest line, 

intersecting it at n= 0 and at n —■—. 

m 



Fig. 1. 

PROBLEMS 

1. Using logarithms, find the amount of $100 for 5 years, com¬ 
pounded annually, at 4 per cent. How many places should be given in 
the tables used, if the result is to be correct to the nearest cent? Compare 
your answer with the result obtained from Table II. 

2. How long would it take $1 to double at 4 per cent, compounded 
annually? At 8 per cent? 

3. Find the amount of $1000 for 5 years, 8 months, at 5 per cent, com¬ 
pounded semiannually. Compute this amount also by using the tables 
for the 11 whole periods involved and computing simple interest for the 
fractional period at the end. 

4. A father, at the birth of his son, sets aside a sum that will amount to 

$5000 in 21 years. If it earns 4 per cent, compounded semiannually, 
what is the sum? Ans. $2176.52. 

5. If the population of a town in 1920 was 4526, and its annual rate of 
increase during the previous decade was 8 per cent, what will its population 
be in 1925, if this rate continues? 




NOMINAL AND EFFECTIVE RATES OF INTEREST 7 


6. Show how to use the tables to obtain the value of (l+i) n for values 
of n outside the range of the tables. Find the amount of $1, for 35 years, 
compounded quarterly, at 6 per cent per annum. 

7. Using (3), calculate the interest on $1000, at 5 per cent, for 6 months. 
Compare it with the simple interest for the same period. 

8. Draw the graphs showing the amounts, by simple interest and by 
compound interest, for $100 bearing G per cent, compounded annually. 

8. Nominal and effective rates of interest.—When interest 
is compounded more frequently than once a year, the rate of 
interest quoted is called the nominal rate. Thus, if interest is 
at 6 per cent per annum, compounded quarterly, the nominal 
rate is 6 per cent, but the rate for the quarterly conversion 
period is lj per cent. It is customary to indicate a nominal 
rate of interest by the letter j. If interest is converted m times 
a year, the symbol j (m) may be used to designate both the 
nominal rate and the frequency of compounding. Thus, if 
j {2) = 0.04, interest is being compounded at 2 per cent per 
half year. 

The effective rate of interest is the amount earned by each 
unit of capital during one year. It will be denoted by the 
letter i. Thus, if j {m) is the nominal rate of interest, 


\+i=[\+ 3 ^y. 

From (5) one obtains 

i.U+hsY-i 

\ m J 

and 

i 

jmy =m{{l+i) m - l\ 


(5) 

( 6 ) 
( 7 ) 


It should be noted that j a) =i, but that, for m greater than 1, 
% will always be larger than j, m) . Thus, if ju, =0.06, 

f = (1.015) 4 -1=0.061364. 

On the other hand, if i =0.06, Formula (7) gives 

j' (4 ) =4{ (1.06) M —1} =0.058695. 







8 


INTEREST 


The reason for these relations becomes apparent when 
one considers that, when interest is compounded more 
frequently than once a year, the interest, thus added to 
the principal itself earns interest, which increases the annual 
earnings and thus makes the effective rate larger than the 
nominal rate. 

Table VIII gives the values of j (2) , j (4) and j (r2) for rates of 
interest used in the tables of this book. When a nominal rate 
of interest is given, Table II may be used, together with For¬ 
mula (6), in order to determine the corresponding effective rate. 

PROBLEMS 

1. Find the effective rate of interest when money is compounded semi¬ 
annually at 5 per cent. 

2. What nominal rate, interest compounded quarterly, will produce 
an effective rate of 8 per cent? 

3. If a business grows from $12,000 to $16,000 in 5 years, what is the 
effective rate of interest involved? 

4. Given, f = 0.06; compute j (2) , j (4 ), j( 12 ) and j^ 65) to four decimal 
places. 

5. Which is the better investment, one in which interest is at 6 per 
cent, compounded quarterly, or one in which simple interest is earned at 

per cent per annum? Compare the earnings in the two cases for 
one year on $1000 principal. 

9. Continuous compound interest. Force of interest.—In 

the relation (7) 

y 

j(m) =m{ (l+z) m — 1}, 

it was seen that, for a given effective rate, i, j {m ) will diminish 
as m, the number of conversions per year is increased. The 
length of the interest period will then diminish, and, as m 
increases without bound, one obtains a state of affairs in 
which interest may be thought of as being compounded con¬ 
tinuously. The limit that j( m ) approaches as m approaches 


CONTINUOUS COMPOUND INTEREST 


9 


infinity is called the force of interest. It is denoted by the 
letter 8. Hence, 

i_ 

5=lim — 1]. 

m = cc 

Expanding (1 by the binomial theorem, and simplifying, 


8 = lim 

m = cc 


—-l 

m 


— — l) (— — 2 


, m 


m 


2.3 


-t 3 + . . . 


i 2 , i 3 i 4 


1 3 • • • • 


The latter series, however, is the expansion of log e (l + z), 
where e = 2.71828 . . ., the base of the natural system of log¬ 
arithms. Hence, 

<5=log c (1+f),.(8) 

and 

1 +i=e s .(9) 

Example. —Find the force of interest when i = 0.06 (see Problem 4, § 8). 

5 = log e 1.06 

log io 1.06 
logio e 

= 0.05827. 

By Formula (9), the amount, S , is given by 

S = P(l+i) n = Pe nS . 


( 10 ) 


PROBLEMS 

1. The valuation of property in a given community may be thought 
of as increasing continuously. If in 1920 it was $10,000,000, and the 
rate of continuous growth is constant, 5 = 0.05, what will the valuation be 
in 1930? 

2. What is the force of interest corresponding to a nominal ratej( 4 ) = 
0.08? 

3. Find the amount of $1000 for 20 years at 6 per cent, compounded 
annually. What would the amount be if interest were compounded con¬ 
tinuously at 6 per cent (5 = 0.06)? 










10 


INTEREST 


10. Present value.—In § (6) it was seen that P units of 
capital, put at interest now at rate i, will amount to S units in n 
years, where 

£=P(l+z) n . 

This means also that a promise to pay S dollars in n years 
could be discharged equitably by the payment of P dollars now. 
For this reason, P is called the present value of S. The term 
present is used in a technical sense, for one may compute the 
“ present value ” of a sum S, n years before it is due, which 
date may or may not be the “ present ” time for the computer. 
The term, however, may be justified by imagining the present 
to be the date at which the value is to be computed. 

It is customary to represent by the letter v the present value 
of 1 due in one year. Hence, 

or 


so that 


The quantity v is always less than 1; hence, the larger n is, 
the smaller will v n be, and therefore, the more remote the 
date at which S is due, the smaller P will be. This, of course, 
is otherwise obvious, from the nature of the relation between 
P and S. 

The values of powers of v are given in Table III for usual 
rates of interest. 

If interest is compounded m times a year and the corre¬ 
sponding nominal rate j im) is given, then the fundamental 
relation (5) gives 


1 =v(l +z), 


1 


V = 


l+i’ 


• • 


• • • 


. ( 11 ) 


P=S 


1 


(1 +f) n 


~- m =Sv 


• • 


. ( 12 ) 


1 


1 







PRESENT VALUE 


11 


Hence, in terms of j (m) , 


P=S 



(13) 


When j(.m) is given, Table III may still be used to find P, 

k 

if the rate of interest — is one of those listed; otherwise, P 

m 

would have to be found by use of logarithms. 

As an example showing the use of the table, suppose it is 
desired to find the present value of $1000 due in 5 years, where 
interest is compounded quarterly at the nominal rate of 7 
per cent, i.e, 4 > = 0.07. Formula (13) gives 


P = 1000 


1 

(1.0175) 20 ' 


This is the same as finding the present value of $1000 due in 
20 years, interest being allowed at the rate of If per cent. 
From the table, the value of (1.0175) ~ 20 is found to be 
0.7068246; hence, 

P =$1000X0.7068246, 

= $706.82. 


PROBLEMS 

1. Find the present value of $1000 due in 10 years, if money is worth 
6 per cent effective. 

2. Find the present value of $1000 due in 10 years, if money is worth 
6 per cent, converted semiannually. (Note that P = 1000i ,2 ° at 3 per cent.) 

3. What sum should be deposited in a savings bank paying 4 per cent, 
compounded semiannually, in order that in 10 years it may amount to 
$6000? ^ 

4. Which is the better offer for a piece of property: (a) $2000 cash and 
$1000 at the end of each year for 3 years; or (6) $1250 cash and $1250 at 
the end of each year for 3 years? Make a comparison on a 5 per cent 
basis, finding the equivalent present values. 

Ans. (a) $4723.25; (6) $4654.06. 







12 


INTEREST 


5. Find the present value of $10,000 due in 20 years, interest at the 
rate of 8 per cent effective. Find the present value with interest at 4 per 
cent effective. 

6. In Problem 5, what would the present value be if j(4) =0.08? 

11. Discount. —The difference between a sum of money due 
at a future date and its present value is called discount. The 
rate of discount is the discount for one year on one unit of 
principal. If this be denoted by d, then, by the definition, 


d-l-v . 

. . . . (14) 

Replacing v by its value, ^ ., 


7 i 

a= . .—iv. . . . 

l+i 

. . . . (15) 

from which also, 


d 

1 1 -d . 

. . . . (16) 


When the rate of discount is given, it is important to know 

the corresponding rate of interest. This is given by (16); and, 

while it is expressed in terms of effective rates, the same formula 

may be used to give the relation between the rate of interest 

and the corresponding rate of discount, for fractions of a year. 

Thus, if a bank discounts a bill due in 90 days, at the nominal 

rate of 6 per cent per annum, it means that the discount for the 

• 

period is 1| per cent. The corresponding rate of interest is'-p 
Formula (16) gives, therefore, 


j( 4) 

4 


0.015 

0.985 


= 0.01523. 


The corresponding effective rate of interest is given by the 
fundamental relation 











PRINCIPLE OF EQUIVALENCE 


13 


from which i — 0.0623. Hence, a person who uses his funds to 
discount 90-day commercial paper at the nominal rate of 6 
per cent is earning nearly 6} per cent on his money. 

PROBLEMS 

1. A bill for $278.50, due in 90 days, was sold for $270. What was 

the nominal rate of discount? Ans. 12.2 per cent. 

2. A salary check for $300, due on July 1, was discounted June 20 at 
7 per cent. How much was deducted? 

3. Paper due in 60 days is discounted at the nominal rate of 8 per cent. 

What is the corresponding nominal rate of interest? Ans. 8.1 per cent. 

4. Find the effective rate of interest corresponding to a discount rate of 
2 per cent per quarter. 

5. Derive Formulas (14) and (15) by considering an amount v, invested 
now and earning d in one year, as interest. 

12. Principle of equivalence. —In this chapter it has been 
made clear that time is an important factor in determining the 
amount of money necessary to discharge a debt. The sum 
that will clear an obligation to-day will not be sufficient a 
year from now, or at any other subsequent date. The dif¬ 
ference is due to interest, and interest has been seen to be an 
increasing function of the time. Thus, if money is worth 
6 per cent, and no interest is paid in the meantime, a debt of 
$1000 to-day can be cleared five years from now only by the 
payment of 

$1000(1.06) 5 =$1338.23. 

Under the assumed interest rate, it may be said that $1000 
to-day is equivalent to $1338.23 five years hence. The same 
debt could be cleared two years hence by the payment of 
$1123.60; it could have been cleared one year ago by the 
payment of $943.40. All of these sums are equivalent. 

As a further illustration, suppose that one debt is to be dis¬ 
charged by the payment of $1000 one year hence, and another 
by the payment of $1500 in two years. What amount could 


14 


INTEREST 


be paid at the end of eighteen months to discharge both obliga¬ 
tions equitably, if money is worth 5 per cent? 

This problem can be solved by letting X be the required 
payment and equating it to the sum of the equivalent values 
of the two debts. Thus, 

X =$1000(1.05)^+11500(1.05)-* 

= $1024.70+$1463.85 
= $2488.55. 

The same result would have been obtained had the equiv¬ 
alent sums been computed at any other date and the corre¬ 
sponding equation formed. Thus, if the present values had 
been found, the result would have been 

X# = 1000^+1500+, 
or, 

X = 1000^+1500+*, 

which gives the same value as found above. 

In the succeeding chapters, the principle of equivalence of 
values, under the compound interest law, will be fundamental. 
This principle is, indeed, directly or indirectly involved in 
nearly every financial problem. 

MISCELLANEOUS PROBLEMS 

1. Find the exact simple interest on $200 for 73 days at 7 per cent. 
Compare it with the interest as found by using (3), putting n=0.2. 

Ans. $2.80; $2.73. 

2. How long will it take $1 to double itself at 6 per cent, compounded 
annually? How long if compounded quarterly? 

3. Find the amount of $1 at 4 per cent interest, compounded semi¬ 
annually, for 100 years. 

4. Construct a graph showing the amount, when $1 bears interest at 
5 per cent effective. Calculate the amounts for every half year for 5 years. 

5. What is the effective rate of interest when 1 per cent per month is 
charged (j(i 2 ) =0.12)? 


MISCELLANEOUS PROBLEMS 


15 


6. Find the force of interest corresponding to an effective rate of 
4 per cent. 

7. Find the nominal rate of interest realized, if a bill for $500, due in 
90 days, is discounted for $490. 

8. If funds are utilized to discount 60-day paper at 6 per cent nominal, 
what effective rate of interest is realized? 

9. If money is worth 7 per cent, what sum, paid one year hence, will 
equitably discharge two obligations, one due in 9 months for $250, the 
other due in 18 months for $400, each without interest. 

10. An obligation is to be discharged 3 years hence, by the payment 

of $3000. Find the amount of each of two equal payments, one to be 
made 1 year hence and the other 2 years hence, that will be the equivalent, 
if money is worth 6 per cent. Ans. $1373.88. 

11. If $300 is due in 30 days, $250 in 90 days, and $600 in 180 days, all 
sums without interest, at what time could the total, $1150, be paid in 
one sum to discharge these debts equitably, money being worth 6 per cent 


CHAPTER II 


ANNUITIES 

13. Definition.— A series of equal payments, made at equal 
intervals of time, is called an annuity. The word implies 
yearly payments, but the term is used to describe any series of 
payments, made at equal intervals of time, which may be of 
any length. Unless otherwise stated, the payments are under¬ 
stood to be made at the end of each interval and to continue 
for a specified number of periods. 

14. Notation. —Given any transaction involving equal 
periodic payments, two important questions immediately arise: 
to find, under given interest conditions, (1) the present value 
of all the payments, and (2) the amount of all the payments 
accumulated to the end of the last period. These computa¬ 
tions are based on an annuity whose total annual payment is 1. 

The following symbols are used: denotes the present 

value of an annuity of 1 per annum for n years, the total 
payment of 1 being made in one installment at the end of each 
year. 

denotes the present value of an annuity of 1 per year 
for n years, the annual payment, however, being made in p 

1 

equal installments of ~, at the end of each pth part of a year. 

denotes the amount of an annuity of 1 per annum for 
n years, and the amount when the annual payment is 
made in p equal installments, at the end of each pth part of a 
year. 

15. Present value of an annuity. —By the definition, 

cin\=v+v 2 -\-c*-\r . • . +y w , 

16 


(1) 


PRESENT VALUE OF AN ANNUITY 


17 


being merely the present value of each payment of 1 due at the 
end of each of the n years. The right member of (1) is a 
geometrical progression whose common ratio is v. Its sum is, 
therefore, 

_v-v n+l 1 — v n 

V 

But- = l+z: hence, 
v 

1-v 11 

a n\= — .. • (2) 

If the annual payment is R instead of 1, and A denotes its 
present value, the formula becomes 

A=R-a-=R^ .(3) 

Table IV gives the values of a ^ for ordinary rates of interest. 

Formula (2) could also be obtained by direct reasoning, in 
the following manner. Suppose SI to be loaned for n years, 
at rate i. The lender is entitled to interest, i, each year and 
to the return of the original dollar at the end of n years. The 
interest constitutes an annuity of i, whose present value is 
ia while the present value of 1 due in n years is v n . Hence, 
the original investment of 1 must provide for ia^, to take care 
of the annual interest, and for v n to be set aside to accumulate 
to 1 in ft years, so as to return the original capital. Hence, 

1 =ia^+v n , 

or, * 

1 -v n 


For values of i, or n, not in the table, a^ must be computed 
by finding v n by means of logarithms, and then performing the 
indicated arithmetical operations. Formula (3) contains four 
quantities, A, R, i, and n. If three are given, the fourth may 
be determined. Except when i is the unknown, this offers 
little difficulty. 









18 


ANNUITIES 


Example 1.—Find the cost of an annuity of $100 per year for 12 years, 
allowing interest at 5^ per cent. This rate of interest is not found in the 
table; hence a must be computed directly. 


1 

~ (1.055) 12 
r?T 2i “ 0.055 


By using logarithms, (1.055) 12 = 0.52601. Substituting and simplifying, 
we find =8.6180. Hence the cost of an annuity of $100 per year is 
$861.80. 

Example 2.—If $10,000 is paid for an annuity yielding $800 per year, 
how many years will it run, if interest is allowed at 5 per cent? 

Formula (3) gives 


Hence, 

Therefore, 


10,000 = 800a-. 

’ n\ 


a— = 12.5 (at 5 per cent). 


l-v n 

.05 


= 12.5 


Therefore, 


v n = 0.375 where v = (1.05) h 


n 


-log 0.375 
log 1.05 


= 20.10 years. 


As defined, a— s requires that n be an integer. The result here, however, 
may be interpreted as indicating that 20 payments of $800 may be made, 
but the payment at the end of the twenty-first year will be less than $800, 
to close the transaction equitably. The cost of an annuity of $800, to run 
20 years at 5 per cent, is 

$SOOa 2 o] =$ 9969 - 76 • 

The difference between $10,000 and this sum is $30.24, which, accumu¬ 
lated to the end of the twenty-first year, amounts to 

$30.24(1.05) 21 = $84.25, 

which is therefore the sum to be paid at the end of the twenty-first year. 

The time could also be obtained directly from the annuity tables, by 
noting that, at 5 per cent, a^ — 12.462, and a^-j = 12.821; hence, the 
value of n that satisfies the equation, a—= 12.5, lies between 20 and 21. 
Indeed, by interpolation, n is found to be 12.106. 






PROBLEMS 


19 


PROBLEMS 

1. What is the present value of an annuity of $100, payable at 
the end of each year for 10 years, if money is worth 6 per cent? Verify, 
by direct computation, the value as found from the tables. 

Ans. $736.01. 

2. Find the costs of annuities of $100, to run 15 years, payable in 
single annual installments, interest being allowed at the following rates, 
respectively, (a) 4 per cent; (6) 6 per cent; (c) 8 per cent. 

Ans. (a) $1111.84; (6) $971.22 (c) $855.95. 

3. A man purchases a house, paying $4000 down and $600 at the end 
of each year for 5 years. What would be the equivalent price if he paid 
all in cash at the time of purchase, money being worth 7 per cent? 

Ans. $6460.11. 

4. A piece of property is purchased for $25,000, the purchaser paying 
$5000 down and agreeing to pay the balance, with interest at 7 per cent, in 
annual installments of $2500. How long will it take to clear the transac¬ 
tion, and how large will the last payment be? 

Ans. 13 years; $345.27. 

5. How much should be paid for a mine that can be made to yield 

$15,000 net per year for 10 years, after which it will be worthless. The 
income is supposed to be available at the end of each year, and the invest¬ 
ment is to yield 8 per cent to the investor. Ans. $100,651.22. 

6. What is the present value of an annuity of $1000, to run 8 years, 
interest being allowed at 7\ per cent? 

16. Formula fora^ } .—In this case the annual payment of 1 

is made in p equal installments of ^ each. Suppose that 

interest be converted in agreement with these payments and at 
the nominal rate j ip) ; then formula (3) may be applied where a 

periodic payment of * is made for np periods, interest being 

. Hence, 

= l a — y^atrate^j. . . . 


allowed at the rate of — per period 

V 


. • ( 4 ) 




20 


ANNUITIES 


This may be expressed in terms of the effective rate of inter¬ 
est, i, by means of the fundamental relation 

l+t-(l+*f) • 

Substituting in (4) the value of a^, 


(p) 

«n| =- 


1 - 



-np 


3(p) 

V 


Hence, finally 


j(p) 


(p) 1 — v n i 

d n \ ■ ~ n w i. 

J (p) J (p) 



Formula (5) should be used when i is given. The factor -— 

J(p) 

can be obtained from Table IX for the usual values of i, and 
for p= 2, 4 and 12, corresponding to semi-annual, quarterly 
and monthly payments. 


PROBLEMS 

1. Find the cost of an annuity of $1000 per year, to run 20 years, 
(a) if payable in one installment and i = 0.04; (6) if payable in two in¬ 
stallments and the corresponding j (2 ) =0.04; (c) if payable in four install¬ 
ments and the corresponding j (4) =0.04. 

Ans. (a) $13,590.33; (5) $13,677.19; (c) $13,722.05. 

2. Find the cost of an annuity of $400, payable in quarterly install¬ 
ments of $100, to run 8 years, interest at 8 per cent nominal (j( 4 > =0.08). 

Ans. $2346.83. 

3. What would the cost be in Problem 2, if interest were at 8 per cent 

effective? Ans. $2366.49. 

4. Find the cost of an annuity of $100 per month, to run 15 years, 
interest at 4 per cent effective. 

5. What is the present value of an annuity that pays $1000 every 
quarter, to run 12 years, interest at 5 per cent effective? 








PROBLEMS 


21 


6. Find the present value of an annuity paying $50 per month for 20 
years, interest at 3| per cent effective. 

7. What is the cost of an annuity paying $600 each half year for 
10 years, interest at 4 per cent nominal OV-o =0.04)? What would the 
cost be if interest were at 4 per cent effective (i = 0.04)? 

17. Formulas for s- { and s - n f .—From the definitions (§ 14) 
of and s^ ;) , it is seen that they are merely the values of the 
series of payments accumulated to the end of the n years. 
But a,n | and represent the values of these same sums at the 


beginning of the n years; hence, 

=Cln\(l+i) n , .(6) 

4'=af(l +i)«.(7) 


Substituting the values of a^j and of ajjf 
ively, 



and 



(l+i)»-l 


3 ip) 


in (6) and (7) respect- 

.( 8 ) 

.(9) 


For purposes of computation, (9) may be transformed as follows: 



(1-H)”-1 i_ 


^ 3(v) 


= SST-r-.(10) 

J(p) 

Table V gives the values of for the usual values of i and n, 

and Table IX gives the values of — for p = 2, 4, 12. 

J(p) 

When the annual payment is R instead of 1, the amount 
S is given by the formulas 

S=R Si j, or S=RC .(11) 

When the payments are made in p installments per year, 
and a nominal rate of interest, j (p) , is given, interest being com- 













22 


ANNUITIES 


pounded in agreement with payments, then the problem 
becomes one of finding the amount of an annuity of - per period, 

“9 / \ 

for up periods, at rate Hence, 

4f = ^ ^at rate .(12) 


Formula (8) can also be derived by the following reasoning 
An investment of 1, with its accumulations of interest, amounts 
to (l+f) n in n years. The annual interest, however, con¬ 
stitutes an annuity, which amounts to fs n - at the end of n years. 
This, with the original dollar, then equals (l+f) M , i.e., 


hence, 


= (1 -\~i) n ] 

”»i 


PROBLEMS 

1. If $1000 is invested at the end of each year for 20 years, at 4 per cent, 
find the amount at the end of the period. 

Find the amount if $500 is invested at the end of every 6 months for 
20 years, interest being allowed at 4 per cent nominal 0\ 2 ) =0.04). 

2. Compare the amounts of annuities of $100 at 5 per cent, running, 
respectively, 5, 10, 15 and 20 years. 

3. Compare the amounts of annuities running 10 years, each for 
$100, at 4 per cent, 6 per cent and 8 per cent, respectively. 

4. Use the s^j table to find how long it will take a man to accumulate 
$10,000, by putting $300 in a savings bank every 6 months, interest at 

4 per cent nominal, converted semiannually. 

Ans. At the end of 13 years amount is $10,101.27. 

5. Find the amount of an annuity paying $150 per quarter, accumulated 
12 years at 6 per cent effective. 

6. If $100 is placed in a savings bank at the end of each month, for 

5 years, and interest is allowed at 4 per cent effective, how much will be on 
deposit at the end of the period? 





DEFERRED ANNUITIES 


23 


7. Find the amount of an annuity of $250 per quarter at the end of 
5 years, at 5 per cent effective. 

8. Find the amount of an annuity of $25 per month for 15 years, interest 
at 6 per cent nominal 0(i2> =0.06). What would the amount be if the rate 
of interest were 6 per cent effective (i = 0.06)? 


18. Deferred annuities. —A deferred annuity is one whose 
payments do not begin until after a certain period of years has 
elapsed. 

The symbols for the present value of an annuity of 1, 
deferred m years, are m\a^ and m\a ( n f , according as the pay¬ 
ments are made once, or p times a year. 

The amounts of deferred annuities, after they have run n 
years, are clearly the same as the amounts of ordinary annuities 
for n years. 

The present value of an annuity deferred m years is the 
same as the present value of an annuity to run m-\-n years, 
diminished by the cost of an annuity for m years. Hence, 


ni\a n \ Q , m + n\ 


I (p) ( p ) ( p) 

Ifll fl = a — - n — 

" l l u ra' u m + n\ u m | ' 


(13) 


These formulas are adapted to computation, the values of 
the quantities in the right members being readily obtained by 
use of the tables. 

It should be noted that another expression for m 
given by 


&n\ 


IS 


m 


m 

^n\ ^ ^ n\J 


because represents the value of the annuity when the 
payments are to begin, and v m . a„\ is its present value. 

19. Annuities due. —When the payments of an annuity are 
made at the beginning of each period, instead of at the end, 
it is called an annuity due. The present value of an annuity 
due of 1 is indicated by the symbol a^, and its amount by s^. 
If the payments are made p times a year the respective sjun- 
bols are a£f and s^f. 

Aside from the first payment of 1, the annuity due, to run 







24 


ANNUITIES 


n years, is equivalent to an ordinary annuity to run n— 1 
years. Hence, 

3-711 = 1 1 |* . 


Similarly, 



1 . (p) 

= -+a 

V 


i 

n - 

V 



The amount of an annuity due, s^, may be found by con¬ 
sidering each payment carried forward to the end of its period, 
when it will amount to 1+i. The annuity due is then equiv¬ 
alent to an ordinary annuity of 1+f per annum, running n 
years. Hence, 

s^=(l+z)^j.(16) 

Similarly, 

s hf=(l+^4f.(17) 


One can also think of as the amount of an ordinary annu¬ 
ity running w+1 years, with the last payment omitted, so that 

s n| =s n+TI “ 1.(18) 

Similarly, 



= s 


(p) 

n +1 


V 


1 

V 


(19) 


PROBLEMS 

1. What sum of money should be set aside now in order to provide 

$1200 a year for 4 years, the first payment to be made 18 years hence, 
interest at 4 per cent effective? Ans. $2236.19. 

2. Find the cost of an annuity of $100 per month, deferred 10 years 
and to run 8 years, interest at 5 per cent effective. Ans. $4869.50. 

3. How much money should be set aside on Jan. 1, 1923, in order to 
accumulate to a sufficient amount to provide an annuity of $1200, payable 
in quarterly installments of $300, the first to be paid on April 1, 1930, 
the last on Jan. 1, 1935, all sums to bear interest at a rate j( 4 > =0.04? 

4. One thousand dollars is put in a savings bank on Jan. 1, 1920, and 
a like sum every 6 months thereafter until July 1, 1930, inclusive. If 
interest is allowed at 4 per cent, compounded semiannually, how much 
will be on deposit after the last payment? 












THE ANNUITY THAT 1 WILL PURCHASE 


25 


5. One hundred dollars is placed in a savings bank at the beginning 
of each month for 6 years. Simple interest on all balances is allowed 
at 4 per cent, but this is compounded semiannually. Show that this is 
equivalent to an annuity of $607 at 2 per cent, running 12 years. Find 
the accumulated amount. 

6. Prove that a^j = (1 

7. Find the cost of an annuity due, the annual payment of which is 
$400, to run 12 years, interest at 5 per cent effective. 

8. Find the cost of an annuity due, paying $100 per quarter, to run 
12 years, interest at 5 per cent nominal 0‘( 4 ) =0.05). 

20. The annuity that 1 will purchase. —The annual income, 
R, from an annuity whose present value is A, is found by 
solving (3) for R, giving 

R = — .(20) 

an j 


If, in a particular case, we let A = 1, we have 


R=— } 

dn\ 


( 21 ) 


as the income from an annuity that 1 will purchase. 

If the annuity is payable p times a year, then the income 
from an investment of 1 is 


1 = j(P) _ 1 

* a-„{ 


( 22 ) 


The values of ■— can be found from Table VI for ordinary 

U n | 

rates of interest. If the annuity is payable in p installments, 
and the effective rate, i, is given, formula (22) will be used. 
The value of j {v) can be found from Table VIII for p = 2, 4 and 12. 
If j {v) is given instead of i, then from (4) we have 


1 _ V 

ri (P) r\ - 

n] Unp \ 


^at rate . 


(23) 


Thus, the income from an annuity costing 1 is obtained; the 
annuity costing A will produce A times as much. 








26 


ANNUITIES 


21. The annuity that will amount to 1.—The annual pay¬ 
ment, R, necessary to give an annuity that will accumulate to 1 
in n years, is found by putting £= 1 in (11), giving 


B= b 

&n\ 


or R 


q (p)> 
t- »i 


. . . (24) 


according as the annuity is payable in one installment, or in p 
installments, respectively. Since, from (12) and (10), 


and 

s (?> 

Ml 

l 

“ P S?l P\ 

then, 

o(P) 

S Si 

i 

=~ — s«i 


1 

- v 


S<?> 

Ml 

&np\ 

or, 

l 

3(v ) 1 


at rate • 


q(Z^) /) Q 

•\7j 1 s »l 


(25) 

(26) 


The values of — may be obtained from the tables for 
by means of the simple relation 


i 

U-n\ 


b = b +i .( 27 ) 

This may be established by direct reasoning. An investment 
of 1 produces an annuity whose annual yield is —. On the 

@n\ 

other hand, from an investment of 1, there should be the annual 

interest i and in addition a sufficient sum — which, set aside 

annually, will amount to 1 at the end of n years, thus returning 
the original capital. 

Hence, — may be obtained from the corresponding value of 








THE ANNUITY THAT WILL AMOUNT TO 1 


27 


— in Table VI, by merely subtracting from it the rate of 

a n\ 

interest. 

By an analogous argument, 


a ( p) s (p) . 

n| n| 

Example 1.—Find the annual yield of a 10-year annuity payable in 
quarterly installments, interest at 4 per cent nominal, (. 7 ( 4 ) =0.04), pur¬ 
chased for $10,000. 

From (23) the annual yield is 


# = $ 10,000 —=$40,000 - (at 1 per cent). 


a 


From Table VI, 


10 | 

1 


thence 


« 40 | 

= 0.0304556, 

O40| 

# = $1218.22. 


Example 2.—How much should be paid annually in order that the 
accumulation in 10 years may be $2000, interest at 5 per cent effective? 
From (27), 

1 1 

— = -—-0.05 
s 10| «10| 

= 0.0795046, 


whence, 


1 

# = $2000-= $159.01. 

Sl0| 


PROBLEMS 

1. A house is purchased for $15,000, and it is arranged that $5000 
cash be paid, and the balance in 10 equal annual installments, including 
interest at 6 per cent. Find the annual payment. 

2. A debt of $3000 is to be paid off by 36 equal monthly installments, 
including interest at 5 per cent effective. What is the monthly payment? 

3. What sum, invested every 6 months at 4 per cent, compounded semi¬ 
annually, will amount to $5000 in 10 years? 









28 


ANNUITIES 


4 . If j( 4 )= 0 . 06 , what is the quarterly payment necessary to accu¬ 
mulate to $3000 in 5 years? 

1 1 . 

5. Prove algebraically that ~ = ~+ 2 '- 

a n\ 

22. Perpetuities and capitalization.—When an annuity is 

continued for an unlimited period, it is called a perpetuity. 

The present value of a perpetuity of 1 is clearly -, because 

7 / 

this is the amount of capital necessary to produce 1 per annum 
as interest. 

This also may be deduced from the relation 

l-v n 

u n \ — • j 


by letting n increase. For, since v is less than 1, the limit of 
v n , as n increases without bound, is zero. 

Another form of perpetuity occurs when regular payments 
have to be made for an indefinite period, but at intervals of 
several years. For example, a certain part of a plant may have 
to be renewed every k years. A question would arise as to how 
much capital should be set aside, in order to provide, through 
its interest earnings, funds sufficient to pay for these renewal 
charges. If x denotes this capital, then xi would be the annual 
interest, and if S is the amount to be raised every k years, then, 


or, 


xi — S, 


S 1 

x =- . 


(29) 


The quantity given by (29), when added to the first cost, S, 
is called the capitalized cost of the article. 

It is clear that it may also be found by summing the series 


• > 


S+Sv t +Sv 2t +Sv>*+ . . 




PERPETUITIES AND CAPITALIZATION 


29 


an infinite geometrical progression whose sum is 


But, from (27), 


S _S 1 
1— v* i at\ 









lienee, the right member of (30) may be replaced by 


(30) 


% 


-S+U, 

I Sk\ 


which is the first cost plus the present value of an indefinite 
number of renewals. 

For purposes of computation, Formula (30) should be used. 
Formula (29) may also be obtained by reasoning as follows: 
The capital, x, that is to be set aside to provide for the renewals, 
will, in k years, amount to x(l+f) fc . At that time, a sum S 
is to be withdrawn, after which the original capital, x, should 
remain, to be allowed to accumulate for another k years. 
Hence, 

£(1+?-)*— S =X , 

or 

S SI 
X (l+f)*-l i sii 

Example. —Compare the capitalized costs of two machines, on a 
6 per cent basis, one costing $2500 and lasting 5 years, the other costing 
$4000 but good for 9 years. If both are capable of doing the same work, 
which is the better investment? 


$2500 1 

0.06 (Z 5 ] 


= $9891.52. 


A- 

The capitalized cost of the second is 


$4000 1 

0.06 agj 


= $9801.48. 


The latter, therefore, offers a slight advantage over the former. 







30 


ANNUITIES 


MISCELLANEOUS PROBLEMS 

1. A man owes $1000, and is to pay it in monthly installments of $20, 
with interest at 6 per cent nominal 0’(i 2 ) =0.06). How long will it take? 
How much is due just after the last full payment of $20 is made? 

2. Ten thousand dollars is invested in an annuity of $60 per month, 
interest at 6 per cent effective. How long does it run? 

3. If $1000 is placed in a fund at the end of each year, interest at 7 
per cent effective, what will it amount to in 7 years, 11 months? 

4. If $100 is annually placed in a fund drawing interest at 5 per cent 
effective, how long will it have to run before it will be sufficient to buy an 
annuity of $1000 per year for 10 years? 

5. Find the cost of an annuity of $750, deferred 12 years, interest at 
6 per cent effective. 

6 . Find the cost of an annuity of $200 per year, payable in quarterly 
installments, the rate of interest being 5 per cent effective, the payments to 
run 15 years. 

7. A man pays $6000 for a mine; he sets aside $1500 at the beginning 
of each year, for 3 years, for development work, and, at the beginning of 
the fourth year, $3000 for a mill. How much should the mine produce 
annually, beginning with the fifth year, in order to net him 8 per cent 
effective on the whole investment, if the value of the mine is exhausted 
at the end of the tenth year? 

8 . Two hundred dollars per month is put in a savings bank on the first 
of each month, beginning January, 1920. Simple interest at 4 per cent is 
allowed on all balances on deposit, and the accumulated interest is added 
to the principal on July 1 and January 1 of each year. How much will be 
on deposit July 2, 1925? 

9. If it costs $1500 a year to maintain a certain number of dirt tennis 
courts, how much could be spent to cover them with asphalt, to be equiva¬ 
lent, if upkeep is thus reduced to $300 per year, interest at 6 per cent? 

10. Prove algebraically that 

11 

a (p) ^ (p) ' 

n\ n\ 

11. What single payment, made in advance, is equivalent to $100 paid 
at the end of each month for 12 months, if money is worth 6 per cent 
effective? 




MISCELLANEOUS PROBLEMS 


31 


12 . How much must be paid now for an annuity of $250 per quarter, 

the first payment to be made 5| years hence, interest at the =0 05, 

the payments to terminate after 20 have been made? 

13. Compute the value of by regarding it as a sum of money drawing 
interest at rate i, compounded annually, but from which 1 is withdrawn 
«at the end of each year for n years, at which time the fund is exhausted. 

14. How long would it take to pay off a debt of $800 by making monthly 
payments of $ 20 , allowing interest at 6 per cent nominal, i.e., ,712 = 0.06? 
How large is the last payment? 

15. If a nominal rate j( m ) be given instead of j^, then, since from (7), 


and since 


Hence, from (5), 


kv) = V Kl+i)*-1), 


m 


X+i^d+^l , 


m 


m 


*» =p U 1+ ^r) - 1 


i-i 


J (m) 


— mn 


a& = 

n 


m 


V 


m 

j (m) \ P 




-1 


16. Use the result of Problem 15 to obtain the cost of an annuity of 
$800, payable in quarterly installments for 10 years, 0' 2 = 0.06). 

17. Obtain the formula for a^ by finding the present value of each of 

the np payments of —, and summing the resulting geometrical progression. 

V 

18. Prove, by direct reasoning, that 

m\an\=Sn\-V m + n =(ln\-V m . 

19. From the tables, find approximately the rate of interest, if an 
annuity of $100 amounts to $3492.58 in 20 years. 

20. Find the formula for by finding the amount of each payment at 
the end of n years at rate i, and adding them. The sum is a geometrical 
progression. 











32 


ANNUITIES 


21. When a rate of interest j(m) j is given, show that 



(See problem 15.) 

22. Use the result of Problem 21 to find the amount of an annuity of 
$250, paid in quarterly installments of $62.50 each, running 8 years, inter¬ 
est being allowed at 4 per cent, converted semiannually. 






CHAPTER III 


AMORTIZATION—SINKING FUNDS 

23. Amortization.—The extinction of debts by uniform 
periodic payments occurs frequently in financial transactions, 
and gives rise to an important application of annuities. 

If K represents a debt, then the annual payment, R, 
necessary to extinguish K in n years and to pay interest 
charges, is the annual payment of an annuity that K will buy. 
Hence, 

R=K~ .(1) 

«nl 

The general process by which the principal of a debt is 
repaid by periodic payments is called amortization. The term, 
however, will, in this chapter, be limited to the method just 
described, whereby the debtor makes equal periodic payments, 
which include both interest and a partial return of principal. 
The interest charge decreases as the principal is reduced; con¬ 
sequently, as time goes on, an increasing amount of the fixed 
periodic payment is applied to the reduction of principal. 
This fact is illustrated in the next article. 

24. Amortization schedules.—Consider a debt of .$1000 
bearing 6 per cent interest. Suppose that it is desired to 
repay this in 10 equal annual installments, including interest. 

From (1) the annual payment will be 

R =$1000 — (at 6 per cent), 

a io'\ 

= $135.87. 

Interest for the first year will be $60; hence, $75.87 of the first 
payment would be applied to the reduction of principal, leaving 

33 




34 


AMORTIZATION—SINKING FUNDS 


$924.13 due at the beginning of the second year. The interest 
on this amount for a year is $55.45; hence the principal is 
reduced by $80.42 by the second payment. The continuation 
of the process may be readily traced in the following table, 
which may be called the amortization schedule . 


Year. 

Principal at 
Beginning 
of Year. 

Interest at 

6 Per Cent. 

Principal 

Repaid. 

1 

$1000.00 

$60.00 

$ 75.87 

2 

924.13 

55.45 

80.42 

3 

843.71 

50.62 

85.25 

4 

758.46 

45.51 

90.36 

5 

668.10 

40.09 

95.78 

6 

572.32 

34.34 

101.53 

7 

470.79 

28.25 

107.62 

8 

363.17 

21.79 

114.08 

9 

249.09 

14.95 

120.92 

10 

128.17 

7.69 

128.18 



$358.69 

$1000.01 


The total amount paid during the 10 years is $1358.70, 
which checks with the sum of the last two columns. 

25. Amount of unpaid principal.—It is important to know, 
at any time, the amount of unpaid principal. Such informa¬ 
tion is needed in keeping accounts and calculating liabilities, 
and for the purpose of closing the transaction at an earlier 
date. This can be learned, of course, from a schedule such as 
that given in § 24. It can, however, be obtained by simply 
noting that, if k payments have been made, the remaining 
n—k constitute an annuity whose present value is the out¬ 
standing principal. Denoting this by A k , then, 


./±i: Ra n _ £|, 

where R has the value given by (1). 


. ( 2 ) 















SINKING FUNDS 


35 


PROBLEMS 

1. Find the annual payment that will be necessary to amortize, in 5 
years, a debt of $1000, bearing interest at 7 per cent. Ans. $243.89. 

2. A man owes 2000 on an automobile, and wishes to pay it off, with 

interest at 6 per cent nominal, in 15 equal monthly installments (j(n) =0.06). 
How much should he pay monthly? How much will he still owe at the 
end of 1 year? Ans. $138.73; $412.04. 

3. Construct a schedule showing the amortization of a debt of $25,000 
in 10 equal semiannual payments, interest at 8 per cent nominal (j( 2 > = 
0.08). 

4. A debt of $ 2000 , bearing interest at 6 per cent nominal (j(n) =0.06), 
is being paid by equal monthly installments, running 5 years. How much 
will still remain due 2 years hence? 

5. A debt of $8000 is to be paid in 5 equal annual installments, includ¬ 
ing interest at 7 per cent. What is the annual payment, if the first is 
made immediately instead of at the end of the first year? 

Ans. $1823 48. 

6 . A person owes $8000. He arranges to pay it, principal and interest, 
in 12 equal semiannual installments, interest at 6 per cent nominal 
0( 2 ) =0.06). After 8 payments have been made, a new arrangement 
is agreed upon, whereby the balance is paid in 6 additional equal payments, 
instead of 4. Find the amount of each of the latter payments. 

7. A debt of $2500, with interest at 6 per cent nominal 0 '(i 2 ) =0.06), 
is being paid off in 30 equal monthly payments. At the end of 2 years, the 
debtor wishes to pay the balance in cash. How much should he pay, 
including the last monthly payment? 

26. Sinking funds.—When an obligation becomes due at 
some future date, it is frequently desirable to anticipate the 
necessary payment by accumulating a fund by periodic con¬ 
tributions, together with interest earnings. This is called a 
sinking fund. 

For example, a corporation issues $1,000,000 in 6 per cent 
bonds, due in 15 years, interest payable semiannually. They 
pay the $30,000 interest charge every 6 months, but, in addi¬ 
tion, wish to set aside, semiannually, a sum sufficient to accumu¬ 
late to $1,000,000 in 15 years, at which time they must redeem 
the bonds. Suppose that they can earn only 4 per cent on their 


36 


AMORTIZATION—SINKING FUNDS 


sinking fund, compounded semiannually. From (24), § 21, 
the necessary semiannual payment is seen to be 

R = $1,000,000-^, (at 2 per cent), 

s 30| 

= $24,649.90. 

The total semiannual payment necessary to take care of this 
debt, principal and interest, is therefore, $54,649.90. 

If they were to accumulate their sinking fund at 6 per cent, 
converted semiannually, the total semiannual charge, including 
bond interest, would be $51,019.30. This is the same as the 
amortization charge, as given by (1). 

In general terms, the annual payment, R, into a sinking 
fund which is being accumulated at rate i, and which must 
amount to K in n years, is given by 

R=K-—j (at rate i) .(3) 

If the debt K bears interest at rate i', then the annual in¬ 
terest charge is Ki f . If we denote by R' the combined sinking 
fund and interest payments, we have 

R'^K-^+Ki'. ...... (4) 

Sn 

But, from (27), § 21, we have, 

-_~-i 

Snf Cln\ 

Hence, 

R'=K —--} -K(i' —i). .(5) 

dn\ 

If the sinking fund accumulates interest at the same rate as 
that paid on the debt K, then i' = i and 

dn\ 



AMOUNT IN THE SINKING FUND AT ANY TIME 37 


The total annual payment for interest and sinking fund charge 
is, therefore, the same in this case as by the amortization 
method. This was illustrated in the foregoing example. 

27. Amount in the sinking fund at any time.—If S r repre¬ 
sents the amount in the sinking fund at the end of r years, 
its value can be found by computing the amount of an annuity 
of R per annum, which has run r years. 

If K represents the debt and n the number of years before 
it is due, we have, from (3) 


hence, 

or, 


R=K — 

S»| 

S r — R ■ s r |, 

Sr=K- S ^ .( 6 ) 


PROBLEMS 

1. A debt of $6000, bearing 7 per cent interest, is due in 4 years. A 
sinking fund is to be accumulated at 5 per cent effective. What is the 
annual payment necessary to take care of both interest and sinking fund? 

Ans. $1812.07. 

2. A city has a bonded indebtedness of $1,000,000, maturing in 20 
years. A sinking fund is created, on which 4 per cent is earned, converted 
semiannually. How much will be in the fund at the end of 10 years? 

3. A city with $40,000,000 assessed valuation issues $300,000 worth 

of bonds, redeemable in 25 years and bearing interest at 5 per cent. A 
sinking fund is created, yielding 4 per cent effective, into which equal 
annual payments are made. How much will the tax rate of the city be 
increased to provide interest on the bonds and to pay the sinking fund 
charge? Ans. 5.55 cents on each $100. 

4. What must be the monthly payment into a sinking fund in order to 

accumulate to $5000 in 3 years, interest being allowed at the nominal rate 
.712 = 0,06? Ans. $127.11. 

5. Quarterly payments are being made into a sinking fund on which 
5 per cent interest is earned (j( 4 ) =0.05). How much is the quarterly 
payment if the sinking fund is to amount to $20,000 in 8 years? 



38 


AMORTIZATION—SINKING FUNDS 


6. A debt of $8000 is to be paid off at the end of 6 years, from a sinking 
fund earning interest at 4 per cent nominal, converted semiannually. 
Find the amount of the semiannual payment into this sinking fund if 
made at the beginning of each half year. 

28. Amortization of bonded debt. —If it is desired to repay 
a debt represented by bonds of a given denomination, it is not 
possible to make the annual payments of principal and interest 
exactly equal, because the amount paid for reduction of prin¬ 
cipal must be a multiple of the face value of the bonds. In 
such cases the amount R, necessary to repay the debt in equal 
annual payments, is determined as in § 24. After the interest 
has been deducted from R in any given year, the number of 
bonds that can be retired with the balance may be determined, 
and a schedule constructed. If the bonds are bought in the 
open market, this schedule will have to be carried forward from 
year to year, in order to be accurate, particularly if there is 
considerable fluctuation in the prices at which the bonds are 
bought. 

Example. —Construct a schedule showing the retirement of an indebt¬ 
edness represented by 50 bonds of the face value of $1000, bearing interest 
at 6 per cent, payable annually. The annual payments for principal and 
interest are to be as nearly equal as possible, the whole debt to be repaid 
in 5 years. 

If the annual payments were all equal, each would equal 

1 

R = $50,000 — = $11,869.82. 


Interest for the first year would be $3000. Subtracting this from R 
leaves $8869.82. The number of bonds that should be retired the first 
year is 9, this being the nearest multiple of $1000 that can be used to ap¬ 
proximate $8869.82. The following schedule shows the continuation of 
the process until the end of the 5 years. The total annual payment, 
shown in the last column, is sometimes larger than R and sometimes 
smaller, but differs from it always by less than $500. The number of 
bonds that can be retired each year increases as the interest charge 
diminishes. 


DEPRECIATION 


39 


Year. 

Principal. 

Interest 
at 6 Per 
Cent. 

No. of Bonds 
Retired. 

Principal 

Repaid. 

Total 

Annual 

Payment. 

1 

$50,000 

$3000 

9 

$9000 

$12,000 

2 

41,000 

2460 

9 

9000 

11,460 

3 

32,000 

1920 

10 

10,000 

11,920 

4 

22,000 

1320 

11 

11,000 

12,320 

5 

11,000 

660 

11 

11,000 

11,660 



$9360 

50 

$50,000 

$59,360 


PROBLEMS 

1. Construct a schedule showing the retirement of an indebtedness 
represented by 100 bonds of the denomination of $1000, bearing interest 
at 5 per cent, payable annually. The annual payments for principal and 
interest are to be as nearly equal as possible, the whole debt to be repaid in 
8 years. 

2. Construct a schedule for the retirement in 5 years of a debt repre¬ 
sented by 1000 bonds, each of par value $100 and bearing 4 per cent interest, 
payable semiannually. Suppose that bonds are repurchased in the open 
market at 102. Arrange the schedule so that the amount paid for interest 
semiannually, together with the amount paid for the bonds which are to 
be repurchased semiannually at interest dates, shall be as nearly equal as 
possible. 

29. Depreciation.—In the operation of physical property of 
every kind, there is a deterioration that cannot be provided 
for by current repairs. Buildings, machinery and equipment 
of all sorts diminish in value through use and through the mere 
action of the elements. Buildings may last fifty years or more, 
while wooden piles in salt water survive only a short time, and 
machinery a relatively brief period, depending primarily upon 
use. This loss in value which cannot be made good by current 
repairs is called depreciation. 

It is a fundamental principal of economics that capital 
invested in business enterprises should not be impaired. From 
current revenues, then, there should be set aside sufficient 




















40 


AMORTIZATION—SINKING FUNDS 


amounts to replace worn-out articles, or to keep intact the 
amount of capital originally invested in them. To do this it 
is necessary to know the probable life of the article and to have 
an estimate of its residual, or scrap, value. The accumulation 
of a depreciation fund may then be accomplished by the 
sinking-fund method. A sum can be set aside annually, 
which at the end of the life of the article will amount to the 
difference between the original cost and the scrap value. This 
difference is called the depreciable value, or wearing value. 
At any intermediate date during the life of the article, its 
book value may be taken to be the original cost less the amount 
in the sinking fund; therefore, the two items taken together 
preserve the original capital intact at all times. 

If C is the cost, S the scrap value, and n the estimated life, 
the annual depreciation charge will be given by 

D = (C-S)~ = W-~, .(7) 

o n \ «rai 

where W denotes the wearing value. 

The amount in the depreciation fund at the end of r years 
is Ds- t j. Hence, the book value at that time is 


C-Dsji=C-W S. (8) 

^n| 

Example. —Suppose that an article costing $1200 has a scrap value of 
$200 at the end of 10 years. It is proposed to accumulate a sinking fund 
at 4 per cent to replace the capital lost by depreciation, and to regard the 
book value of the article, at any time during the interval, as equal to the 
original value of $1200 diminished by the amount in the sinking fund. 
Construct a schedule showing the amount in the sinking fund at the end 
of each year and the resulting book value of the article. 

By (3), § 26, the amount that must be paid annually into the sinking 
fund is 

Z> = $1000 — =$83.29. 
s 10[ 

The amount in the sinking fund at the end of any year, r, can be found 
by computing Ds or can be found by calculating the interest on the 


DEPRECIATION 


41 


amounts in the sinking fund year by year and adding it to the sinking fund 
together with the annual payment. The following table shows the results: 


Year. 

Book Value 
at Beginning 
of Year. 

Total Amount 
in Sinking Fund 
at End of Yea . 

Interest 

on 

Sinking Fund 

1 

$1200.00 

$ 83.29 

$ 0.00 

2 

1116.71 

169.91 

3.33 

3 

1030.09 

260.00 

6.80 

4 

940.00 

353.69 

10.40 

5 

846.31 

451.13 

14.15 

6 

748.87 

552.47 

18.05 

7 

647.53 

657.86 

22.10 

8 

542.14 

767.46 

26.31 

9 

432.54 

881.45 

30.70 

10 

318.55 

1000.00 

35.26 

11 

200.00 




















42 


AMORTIZATION—SINKING FUNDS 


In Fig. 2 the growth of the sinking fund and the corre¬ 
sponding decrease in book value, in the preceding example, 
are illustrated graphically. 

PROBLEMS 

1. The capital represented by an auto truck costing $1600, with a 
probable life of 8 years and a scrap value of $200, is to be replaced by money 
set aside in a 4 per cent sinking fund by equal annual payments. Find the 
amount of the payment and the book value at the end of 5 years. 

2. A plant consists of three parts, with costs, scrap values, and probable 
lives as given in the following table: 


Part. 

Cost. 

Scrap Value. 

Life. 

A 

$50,000 

$5000 

25 years 

B 

20,000 

3000 

15 years 

C 

10,000 

1000 

8 years 


Find the total annual payment into a sinking fund, accumulated on a 
4 per cent basis that will be necessary in order to provide for depreciation. 
What will the total book value be at the end of 8 years? 

Ans. $2906.29; $53,220.78. 

3. A pipe line has a probable life of 15 years. If its wearing value is 
$100,000, what should the annual depreciation charge be, on a 4 per cent 
basis? Find the book value at the end of 10 years. 

30. Other methods of estimating depreciation.—The sink¬ 
ing-fund method of allowing for depreciation, as just discussed, 
is only one of many plans used in practice. Its chief advantage 
is that the annual depreciation charge is constant. The 
decrease in book value varies, being slightly greater each year 
than it was in the preceding year, and becoming more rapid 
toward the end of the life of the article. 

Another procedure, known as the straight-line method, 
makes the decrease in book value the same each year. Thus, 
in the example discussed in § 29, $100 would be “ written off ” 
each year. If this amount were set aside without interest it 










OTHER METHODS OF ESTIMATING DEPRECIATION 43 


would provide the necessary sum for replacement at the end 
of 10 years. If it is invested during the interval, the interest 
earned can properly be turned back into income. 

Some corporations follow a plan of estimating depreciation 
by allowing a fixed percentage of the total valuation each year. 
The rate is determined, as nearly as possible, so as to provide, 
with interest accumulations, a fund from which sums can be 
drawn to provide for replacements as they become necessary. 
Referring to the example in § 29, let r denote the constant 
percentage to be deducted each year; then the value at the 
beginning of the second year would be $1200(1 —r). At the 
end of the second year it would be $1200(1 —r) 2 and so on. 
If the scrap value at the end of 10 years is $200, then 

1200(1 —r) lo = 200. 

By use of logarithms, r may be found from this equation to be 
equal to 16.405 per cent. 

In the general case, r is determined by the equation 

C(l — r) n = S . (9) 

As in the case of the straight-line method, the amount written 
off each year may be put aside in a sinking fund, and any 
interest earned returned to the general revenue account. 

For other methods of estimating depreciation, the student is 
referred to books on accounting. 

PROBLEMS 

1. If an article, which is worth $2400, when it is new, depreciates in 8 
years to a scrap value of $400, construct a schedule showing its book value 
by the straight-line method of estimating depreciation. 

2. What constant percentage would have to be written off each year if 
that method were used in the preceding example? Construct a schedule 
showing the book value for each year. 

3. A building costing $100,000 is estimated to have a life of 50 years, 
with a residual value of $10,000. Construct and compare schedules show¬ 
ing the book values as computed by the straight-line and by the percentage 
methods. 


44 


AMORTIZATION—SINKING FUNDS 


31. Composite life.—If a business concern contains several 
parts of different probable lives, the composite life is defined as 
the time required for the total depreciation charge to accumulate 
to the total wearing value. Thus, if W\, W 2 , . . W k are the 
wearing values of the individual parts, and m, n 2 , . . . n k , their 
respective duration; then, if 


and 


W = Wi + W 2 + _ +-W k , 

D =Di~\-D2~\- .... -\-D k , 


where D\, Do, . . . D* represents the annual depreciation charge 
for the various parts; then, by the straight-line method 


W 

n - D . 

By the sinking-fund plan 

DSn\ = W, 

or 

W 

® _ D' 


( 10 ) 


The value of n may be determined by the use of logarithms, 
or from the tables by locating the nearest value of in 
the appropriate interest column. Thus, in Problem 2, § 29, 
we have TT = $71,000 and D = $2906.29; whence, 

= 24.43 (at 4 per cent). 

From the tables, n is found to be between 17 and 18 years. 
By interpolation, n= 17.36 years. 

The composite life of a plant is an average of the lives of 
its individual parts, weighted according to their costs, and with 
interest taken into account. Considered in another way, it is 
the life of a hypothetical object whose wearing value is equal 
to the combined wearing values of all the parts of the plant in 
question, and which is being provided for by the annual depre¬ 
ciation payment. The knowledge of its value is useful in 
certain questions arising in finance, such as the determination 



VALUATION OF MINING PROPERTY 


45 


of the term of a bond issue loaned on the property as security. 
Such an issue would ordinarily be for a term not exceeding the 
composite life of the plant, on the principle that money bor¬ 
rowed to purchase equipment, or other adjunct of a business, 
and on these articles as security, at least in part, should be 
repaid before the articles are worn out. 

PROBLEM 

Find the composite life of a plant consisting of the following parts: 


Part. 

Cost. 

Scrap Value. 

Life in Years. 

A 

$100,000 

$10,000 

40 

B 

25,000 

2,000 

25 

C 

15,000 

1,000 

15 

D 

12,000 

500 

7 


(a) Interest at 6 per cent. ( b ) Interest at 4 per cent. 

32. Valuation of mining property.—In operating a mine, a 
quarry, or any similar property in which there is a limited 
product, sooner or later to be exhausted, provision must be 
made to restore, or keep intact, the capital invested in the 
enterprise. This is done by setting aside part of the annual 
income as a redemption fund. 

If the mining engineer is able to make fairly accurate esti¬ 
mates of the total amount of mineral in the mine, and of the 
cost and rate at which it can be removed, the value of the mine 
can be readily computed. Its value is the present value of 
an annuity yielding the estimated net annual revenue, and to 
run the number of years given by the engineer’s estimate. 

Thus, if the annual revenue is R and its life n years, then 
its value V is given by the formula 


V=Rdn\=R 


1 - V n 


( 11 ) 











46 


AMORTIZATION—SINKING FUNDS 


where i is the investment rate. The annual interest on the 
investment is Vi; hence, 

R — Vi=Rv n 

is available out of the annual income as a payment into the 
redemption fund. If this can be accumulated at rate i, it will 
amount in n years to 

Rv n = Ra„\ = V, 

thus restoring the original capital. 

However, on account of certain risks involved in such 
enterprises, the investor usually demands a relatively higher 
investment rate of interest than he can expect to obtain on 
money paid into the redemption fund. If i' denotes the invest¬ 
ment rate, and i the rate at which the redemption fund can be 
accumulated, then the amount annually available for the 
redemption fund is 

R-Vi'. 


If this be accumulated for n years at rate i, and the fund then 
amounts to V, we have 


whence, 


(R-V-i')^ = V; 


V = 


R 


_1 


+i' 


R 



^n\ 


■ ■ ( 12 ) 


where Sn\ is to be computed at rate i. It is to be noted that 
this reduces to equation (11) when i ! =i. 

The foregoing results may also be obtained by using the 
sinking-fund method of estimating depreciation. For if V 
is the total capital to be invested in the property, including 
equipment, and the whole investment is to be abandoned in n 
years, and if D represents the annual payment into a sinking 
fund, sufficient to restore this capital at the end of n years, 
then, by (7), 

D = V~ (at rate i). 

Sni 




MISCELLANEOUS PROBLEMS 


47 


But the amount available for the sinking fund, after allowing 
for interest on the investment, was seen to be 

R-Vi'. 

V- = R-Vi’. 

$n\ 


—+i' 

^n| 

PROBLEMS 

1. Find the value, on a 6 per cent basis, of a mine that can be made to 
yield a net annual income of $20,000 for 15 years. 

2. What will be the value of the mine in Problem 1, if the investment 
rate is 10 per cent, and the redemption fund rate is 5 per cent? 

MISCELLANEOUS PROBLEMS 

1. A man purchases a house for $15,000. He pays $6000 down and 
arranges to pay the balance, with interest at 7 per cent, in 8 equal annual 
installments. Find the annual payment. How much will still be due 
at the end of the fourth year, after his annual payment has been made? 

2. Suppose that, in the preceding problem, the man accumulates a 
sinking fund to meet the payment of $9000 at the end of the 8 years. If 
equal annual payments are made into the sinking fund, and it earns 4 
per cent effective, what will be the total annual payment necessary to take 
care of both interest and sinking fund? 

3. A debt of $12,000, bearing interest at 7 per cent, is being paid off, 
principal and interest, by annual payments of $2000. After four of these 
have been made, the balance then due is to be paid in four additional equal 
annual payments. How large must they be? 

4. A debt of $50,000, with interest at 6 per cent, is to be paid off by 
four equal annual installments, P, followed by four equal annual install¬ 
ments of 2 P. Find the value of P. 

5. For 3 years, $1000 is paid annually into a sinking fund earning 
5 per cent per annum; after which the annual payment is increased to 
$1500, but the rate of interest drops to 4 per cent. If the payments 
f are made at the end of each year, how much will be in the sinking fund at 
the end of 8 years? 


Hence, 

From which, 



48 


AMORTIZATION—SINKING FUNDS 


6. Construct a schedule showing the retirement of 50 bonds of par 
value $1000, and bearing 5 per cent, payable semiannually. The bonds 
are to be taken up at interest payment dates extending over 4 years, the 
schedule to be arranged so that the semiannual payment for interest and 
principal shall be as nearly equal as possible. 

7. An article costing $1800 has a probable life of 8 years, with a residual 
value of $200. Construct a schedule of book values for each year of its 
life, (a) by the sinking-fund method, interest at 5 per cent; ( b ) by the 
straight-line method, (c) by the constant percentage plan. 

8. Find the composite life of a plant consisting of two parts; A having 
a life of 50 years and a wearing value of $60,000, B having a wearing value 
of $20,000 and a probable life of 10 years. Assume that money is worth 
5 per cent. 

9. What constant percentage must be written off each year, if an article 
costing $10,000 is to be reduced to $5000 in 5 years? 

10. If 4 per cent per year is written off on the book value each year, for 
a plant costing $100,000, what should the valuation be at the end of 10 
years? 

11. If a machine has a probable life of 15 years, and, without allowing 
for depreciation, yields an average net return of 10 per cent, what rate of 
income does it produce if a sinking fund is set aside at 5 per cent to replace 
it when it is worn out? 

Ans. 5.366 per cent. 

12. An automobile has a probable life of 6 years, and a wearing value of 
$2000. What is the annual cost to the owner, if his annual charge for 
upkeep and running expenses is $500, and depreciation, on the sinking-fund 
plan, is allowed on a 4 per cent basis? 

13. How much should be paid for a mine that could be made to yield 
$25,000 a year for 12 years, after which it would have to be abandoned, if 
the investment rate is to be 12 per cent, while a redemption fund can be 
accumulated at 5 per cent? 


CHAPTER IV 


BONDS 

33. Description.—The ordinary bond is a promise to pay a 
definite sum on a specified date, and, in the meantime, to pay 
interest on this sum at regular intervals, at a stated rate. 
These intervals are usually half-yearly, but interest may be 
paid quarterly or annually, or in some other regular manner. 

To facilitate the payment of interest, the bond usually has 
coupons attached to it. These are themselves individual 
promises to pay the amount of the interest due at the respective 
interest-payment dates, and are detached and presented for 
payment as they fall due. 

The sum named in the bond is called the face value, or par 
value ; sometimes it is referred to as the denomination of the 
bond. When a bond falls due, and the specified payment of 
principal, and outstanding interest is made, the bond is said to 
be redeemed . Bonds are usually made redeemable at par, but 
sometimes, to make them more attractive to the investor, 
they are made redeemable above par. The bond interest, how¬ 
ever, is always computed on the par, or face value. 

34. The investment rate.—If a bond, redeemable at par, is 
bought at par, the investor will realize a rate of interest on his 
investment equal to the bond rate. Bonds, however, are 
usually bought at prices that yield an investment rate, independ¬ 
ent of the interest rate named in the bond. If it is larger, 
then the purchase price is less than the par value. On the 
other hand, if the investment rate is smaller than the bond 
rate the purchase price is greater than the par value of the bond. 

A number of considerations enter into the determination of 
a proper investment rate for a given bond. Chief among them 

49 


50 


BONDS 


are the value of the security back of the bond, the number of 
years before the bond matures, the marketability of the bond, 
in case the holder wishes to sell at any time, and the prevailing 
interest rates and opportunities for investment. 

Assuming a given investment rate the mathematical theory 
of bonds is primarily concerned with the determination of the 
cost of a bond paying a specified coupon interest and with a 
given number of years to run before it matures. A second 
question is the determination of the investment rate when a 
bond is bought at a certain price. The first of these problems 
is solved directly by the bond formula. 

35. The bond formula.—The following notation will be used: 

F =the face, or par value of the bond, 

C =the redemption price (usually C =F), 
r = the bond rate of interest, 
n =the number of years before redemption, 
p = the number of interest payments per year, 
i = the effective investment rate of interest, 

V n =the value of the bond n years before redemption. 

The value of a bond n years before redemption is made up 
of two parts, (1) the present value of the redemption price, (2) 
the present value of all the interest payments. The latter 
constitute an annuity whose annual payment is rF. The pres¬ 
ent value of such an annuity is rFciff. Combining these two 
quantities, we have 

V n =Cv n +rFa%, .(1) 

where v" and a~\ are to be computed at rate i. 

If, instead of i, the nominal rate jw is given, the interest paid 

tF 

on the bond may be thought of as an annuity of — per period, 

running up periods, interest being compounded at rate —. 

P 

Hence, 

V n = Cv np -\- 1 -^-a^, ^at rate . ... (2) 




THE BOND FORMULA 


51 


It is the practice of bond agencies to quote the price of a 
bond to yield a certain specified investment rate, meaning 
thereby the nominal rate corresponding to the number of 
interest payments per year. Thus, a bond sold at a price to 
yield 6| per cent to the investor means, when interest is pay¬ 
able semiannually, an investment rate of 3| per cent per half 
year (j (2) =0.065). Tables are published giving prices of bonds 
with yearly, semiannual and quarterly interest payments. 
The investment rates, as quoted in these tables, are nominal 
rates, compounding in agreement with interest periods. LTnless 
otherwise indicated, this procedure will be followed in this 
chapter. 

It is also the practice to quote bond prices on the basis of 
100 par value. Thus, a quotation of 98.42 means that a $1000 
bond at that price would sell for $984.20. 

Example. (1) Find the cost of a 15-year 5 per cent bond, redeemable 
at par, interest payable semiannually, bought at a price to yield 6 per cent. 

Here, jp) =0.06 and r = 0.05. The semiannual interest payments on 
$100 par value are $2.50. Hence, from ( 2 ) 

7 i 5 = 100y 30 +2|a— (at 3 per cent). 

From the tables, ✓ 

100y 30 = 41.20, 

2 ^= 49 . 00 . 

Hence, 

Ui5 = 90.20. 

(2) Find the cost of the bond in Example (1) if redeemed at 105. 

In this case, 

V 15 = 105r 30 +2^a30|, (at 3 per cent) 

105y 30 = 43.26, 

2 ^= 49 . 00 . 

Hence, 

Vu = 92.26. 


Unless otherwise stated, it will be understood that bonds are redeemable 
at par. 


52 


BONDS 


PROBLEMS 

1. Find the cost of a 4 per cent $1000 bond, redeemable in 40 years, 
interest payable semiannually, bought at a price to yield 5 per cent. 

Ans. $827.74. 

2. Find the cost of a 5 per cent $1000 bond, redeemable in 40 years, 
interest payable semiannually, bought at a price to yield 4 per cent. 

Ans. $1198.72. 

3. At what price should 5-year 6 per cent bonds be quoted, if redeem¬ 
able at 105, interest payable semiannually, if they are to yield 8 per cent? 

Ans. 95.27. 

4. Compare the costs of two 5 per cent bonds, each paying interest 

semiannually, one maturing in 5 years, the other in 10 years, bought to 
yield \\ per cent. Ans. 102.22; 103.99. 

5. Find the cost of a 6 per cent bond, par value $1000, interest payable 
quarterly, bought at a price to yield 7 per cent, maturing in 10 years 

Ans. $928.51. 

6. Compare the costs of $1000 bonds, paying 6 per cent interest, redeem¬ 
able in 10 years, bought at prices to yield the investor 5 per cent, one paying 
interest annually, a second semiannually, and a third quarterly. 

7. Find the cost of a $1000 bond, redeemable in 10 years at 106, paying 
6 per cent quarterly, bought at a price to yield 7 per cent. 

8. At what price should 20-year, 5 per cent semiannual bonds be quoted 
to yield the purchaser 4f per cent? 

36. Premium and discount.—Further insight into the valua¬ 
tion of bonds may be obtained by calculating the difference 
between the price paid for a bond and its face value. When 
this difference is positive it is called the 'premium ; when nega¬ 
tive it is spoken of as the discount. 

To simplify the discussion, it will be assumed that the bond 
is redeemed at par. Letting C =F in (1) we have 


But, from (5) § 16, 
Hence, 


V n -F=F(v n -l)+rFa$. 

V n ~l= 


V„-F=(r-j {p) )Fa%. 


■ ■ (3) 


PREMIUM AND DISCOUNT 


53 


If interest is considered as compounded in agreement with the 
coupon payments on the bond, then, from (4) § 16 

a n\ = p a nP\- ra t e ' 

Hence (3) becomes 

V *- F = (~f) F -"** .( 4 ) 

These results show, and it is otherwise obvious, that a bond 
sells at a premium when r>j( P) , and at a discount when r<j {p) . 
The premium is seen from (3), or (4) to be equal to the present 
value of an annuity, running n years, whose periodic payment 
is equal to the difference between the interest paid on the bond 
and that required by the investment yield. This result is 
apparent also by direct reasoning, because the cost would equal 
F if the yield rate were the same as the investment rate. The 
excess of the bond interest over the yield requirement, there¬ 
fore, constitutes an additional periodic income, or annuity, 
whose present value the purchaser pays for in the form of the 
premium. 

In the case when the bond interest is less than the yield 
demand, the purchaser may be regarded as having paid F for 
the bond; but the seller gives him a rebate, or discount equal 
to the present value of the deficiency in the coupon payments 
from the amounts that would have been required had the pur¬ 
chase price been F. 

Since the cost of an annuity is greater the longer it has to 
run, so the premium or discount will be greater the longer the 
life of the bond, the other data being the same. 


PROBLEMS 

1. Compute the premium on a $1000 bond, paying 6 per cent, with 20 
years to run, interest payable semiannually, bought at a price to yield 
per cent. What would the premium be if the life of this same bond was 
40 years? 



54 


BONDS 


2. Find the discount on a $100, 4 per cent bond, bought 10 years before 
maturity, interest payable quarterly, yielding 6 per cent to the investor. 
What would the discount be one year before maturity? 

3. An issue of 6 per cent bonds, maturing in 20 years, interest payable 
semiannually is bought at a price to net 5| per cent. Find the premium 
paid. 

37. Amortization of the premium.—When a bond is 
bought at a premium, care should be taken by the investor, to 
conserve all of the capital involved in the purchase. The cap¬ 
ital represented by the face value of the bond is returned to 
him when the bond matures. The capital invested in the pre¬ 
mium, however, is returned as part of the excess of interest 
over yield requirements, as seen in the previous article. The 
amortization of the premium follows the same process, as illus¬ 
trated in § 24. At each interest payment, a portion of the 
premium is returned, and the book value of the bond is “ written 
down” by that amount, gradually approaching par at maturity. 

As an illustration, consider a 6 per cent $1000 bond, redeem¬ 
able at par, January 1, 1923, interest payable on January 1 
and July 1, bought January 1, 1920, at a price to yield 5 per 
cent. The premium is found to be $27.54. From (4), this is 
the present value of an annuity whose semiannual payment is 

-{ r — iw)F =$5.00. 

P 

Proceeding as in § 24, it is found that the interest on $27.54 
for 6 months at 5 per cent is $0.69. Hence, on July 1, 1920, the 
value of the premium has been reduced by $4.31. The book 
value of the bond is, therefore, $1023.23. During the next 6 
months, the interest on the remainder of the premium is $0.58; 
on January 1, 1921, therefore, $4.42 of capital is returned, 
leaving the book value of the bond as $1018.81. This process 
continues until the date of maturity, at which time the entire 
premium has been returned and the value of the bond stands 
at par. 


AMORTIZATION OF THE PREMIUM 


55 


The process of accounting will be found simpler if the pre¬ 
mium be not segregated from the rest of the capital invested. 
Thus, the accountant would charge $1027.54 against this bond 
investment, as of date January 1, 1920. On July 1, 1920, a 
$30 coupon is cashed. The investment at 5 per cent requires, 
however, only $25.69; hence, $4.31 remains for amortization of 
the premium. The continuation of the accounting is shown 
in the following table: 


Date. 

Book Value. 

Semiannual 
Interest at 

Per Cent on 
Book Value. 

Semiannual 
Bond Inter¬ 
est at 3 Per 
Cent on Par. 

For 

Amortiza¬ 

tion. 

Jan. 1, 1920.. . . 
July 1 .. . . 

$1027.54 
1023.23 

$25.69 

$ 30 

$4.31 

Jan. 1, 1921... . 

1018.81 

25.58 

30 

4.42 

July 1 .... 

1014.28 

25.47 

30 

4.53 

Jan. 1, 1922.. . . 

1009.64 

25.36 

30 

4.64 

July 1 .. . . 

1004.88 

25.24 

30 

4.76 

Jan. 1, 1923.. . . 

1000.00 

25.12 

30 

4.88 



$152.46 

$180 

$27.54 


An examination of this table shows that, of the $180 received 
through interest payments, only $152.46 may be regarded as 
income on the investment. The sums in the last column, to 
the amount of $27.54, are return of capital represented by the 
premium. These amounts, as they come in, should be returned 
to the capital account of the owner of the bond, to be again 
invested, but such investment would, be separated from, and 
independent of, the bond investment here considered. 

It should be noted, however, that had the owner of the bond 
reinvested $4.31 each half year at 5 per cent, he could have 
left the investment on his books at the original cost, $1027.54, 
and regarded the $25.69 as a uniform income from the bond. 




















56 


BONDS 


The accumulation of the annuity thus created by the semi¬ 
annual investment of $4.31 would amount to 

$4.31 =$27.54, 

at the date when the bond matured. This amount, together 
with the return of the face value of the bond, would preserve 
the total amount of capital intact. 

PROBLEMS 

1. Construct the schedule showing the amortization of the premium on 
a $10,000 bond, bearing 5 per cent interest payable January 1 and July 1, 
redeemable at par, January 1, 1926, bought July 1, 1922, at a price to yield 
4 per cent. 

2. Construct a schedule for a $1000 bond, bearing 8 per cent interest 
payable quarterly, redeemable in two years at 105, bought at a price to 
yield 6 per cent to the investor. This schedule will show the amortization 
of the excess of the purchase price over $1050, the redemption price. 

38. Accumulation of discount. When a bond is bought at 
a price which is less than the redemption price, its value 
increases as the date of maturity approaches. The bond rate 
of interest is not sufficient, in this case to give the income 
required by the investment. The deficit is the amount by 
which the book value of the bond increases at each interest- 
payment date. This process of “writing up” the book 
value is also called accumulating the discount. This phrase 
does not conform to the definition of discount, except when the 
bond is redeemed at par. 

Example. —Construct a schedule showing the accumulation of the 
discount on a $1000 bond, bearing 4 per cent interest, payable January 1 
and July 1, bought January 1, 1920, at a price to yield 5 per cent, the bond 
redeemable at par January 1, 1923. 

The purchase price is found to be $972.46. Interest on this amount 
for 6 months, at the investment rate, is $24.31. The coupon, however, 
pays only $20. The value of the bond, as computed for this date, would be 
$976.77, which is $4.31 greater than its value 6 months earlier. This is 
just the amount represented by the difference between the yield require¬ 
ment and the bond interest. • The schedule shows the continuation of 
this process to maturity. 


BONDS PURCHASED BETWEEN INTEREST DATES 57 


Date. 

Book Value. 

Semiannual 
Interest at 2 b 
Per Cent on 
Book Value. 

Semiannual 
Bond Interest 
at 2 Per Cent 
Cent on Par. 

Accumula¬ 
tion of 
Discount. 

Jan. 1, 1920 

July 1 

$972.46 
976.77 

$24.31 

$ 20 

$4.31 

Jan. 1, 1921 

981.19 

24.42 

20 

4.42 

July 1 

985.72 

24.53 

20 

4.53 

Jan. 1, 1922 

990.36 

24.64 

20 

4.64 

July 1 

995.12 

24.76 

20 

4.76 

Jan. 1, 1923 

1000.00 

24.88 

20 

4.88 



$147.54 

$120 

$27.54 


PROBLEMS 

1. Construct the schedule showing the accumulation of the discount on 
a $1000 bond bearing 5 per cent interest, payable March 1 and September 1, 
redeemable at par September 1, 1928, bought March 1, 1923, at a price to 
yield 6 per cent. 

2. Construct the schedule for a $1000 bond, redeemable in 2 years 
at 102, bearing interest at 6 per cent, payable quarterly, bought at a price 
to yield 8 per cent. (This schedule will show the increase in the book value 
at interest dates, finally reaching $1020 at maturity.) 

39. Bonds purchased between interest-payment dates.— 

When a bond is purchased between interest-payment dates, the 
seller is clearly entitled to a portion of the interest that has 
accrued since the last coupon was paid. If V is the value of 
the bond at that date, the theoretical value of the bond would 
be V, together with interest on this amount, at the investment 
rate, for the portion of the period that has elapsed. 

Example. —Find the value of a $1000 bond bearing 5 per cent, interest 
payable semiannually, maturing at par January 1, 1927, bought May 1, 
1922, at a price to yield 4| per cent. On January 1, 1922, its value was 
$1022.17. The accumulated amount for 4 months at 4| per cent on this 
sum would be 

$1022.17 (1.0225)** = $1037.45. 

In practice, however, this amount would be computed by simple interest. 



















58 


BONDS 


It is the practice of bond houses and other selling agencies to quote bonds 
at a given price with accrued interest, rather than a price computed on a 
strictly yield basis. This means that simple interest at the rate named in 
the bond is computed for whatever fraction of a period may have elapsed 
since the last interest was paid. Thus, in the foregoing example, if the 
quoted price was SI022.17 and accrued interest, the accrued interest for 
4 months at 5 per cent on the face of the bond, would be S16.67, making 
the purchase price S1038.84. 

Ifi consideration of the fact that the next interest payment is 2 months 
hence, the buyer would be entitled to a discount, amounting to $0.14, 
for advancing the $16.67. This discount is negligible for small transac¬ 
tions, but may become quite appreciable when large sums are involved. 


PROBLEMS 

1. A $100 bond maturing October 1, 1930, bearing 6 per cent interest 
payable April 1 and October 1, is bought August 1, 1923, at a price to yield 
5 per cent to the investor. Find the theoretical value of the bond. 

2. A 5 per cent bond, par value $10,000, maturing January 1, 1927, 
interest payable January 1 and July 1, is sold June 1, 1923, on a 6 per cent 
yield basis. Find its selling price. What would its selling price be if it 
were sold on the basis of its value on January 1, 1922, plus accrued interest? 

3. A $1000 bond, paying 7 per cent semiannually, on April 1 and Octo¬ 
ber 1, was sold on March 1, at 102.25 and accrued interest. What was the 
selling price? 

4. Find the theoretical value of a $100 bond bearing 5 per cent interest, 
payable July 1 and January 1, maturing January 1, 1925, bought June 1, 
1923, on a 6 per cent yield basis. 

40. Calculation of the investment rate when the purchase 
price is given.—The prices of bonds in the open market are 
subject to fluctuations. A purchaser, in determining whether 
a particular bond at the quoted market price offers the kind of 
investment he desires, should know what rate of interest the 
bond will yield. Its determination, then, is of great practical 
importance; but, as is the case with many inverse problems, 
its accurate calculation offers mathematical difficulties. 

An examination of the bond formula, in the form given by 


CALCULATION OF THE INVESTMENT RATE 


59 


shows that the yield rate, i, enters into v and into and when 
V n is given, this equation, expressed in terms of i, is of degree 
n+1. When the value of i, to several decimal places, is desired, 
methods of solution developed in the theory of equations 
may be applied. For practical purposes, however, it is possible 
to determine the approximate yield rate by simple interpola¬ 
tion. In order to illustrate this method it will first be assumed 
that only such tables as are given in this book are available. 



Consider, as an example, a 5 per cent bond maturing in 18| 
years, interest payable semiannually, redeemable at par. 
Figure 3 shows the manner in which the cost diminishes as 
the investment rate varies from 3 per cent to 6 per cent, the 
cost being computed for each .5 per cent interval. The curve 
obtained by joining the points representing the costs is seen to 
be slightly concave upward. Between two adjacent points, 
such as A and B , the curve may be assumed to be a straight 





60 


BONDS 


line, and the yield corresponding to a price falling between the 
prices represented by A and B may be determined by inter¬ 
polation. Thus, suppose the quoted price to be 103.35 and the 
resulting yield x. 


Hence, 


Cost . 
106.23 
103.35 
100.00 


Yield. 

4| per cent 
x per cent 
5 per cent 


2.88 : 6.23 =x —4| : J. 


From which one obtains £ =4.73 per cent. 

This method will always give a result slightly larger than 
the correct answer, because the chord is above the curve, at the 
point where the cost is plotted. The correct answer, in the 
foregoing example, to three decimals, is 4.726 per cent. 

It is obvious that the shorter the chord used the better will 
be the result obtained. The procedure is to estimate about 
where the yield will be and to compute the cost at two rates 
nearest to this estimate, one smaller and the other larger, 
these rates to be determined by rates of interest used in the 
available annuity tables. 

Bond tables may be obtained which give investment rates 
differing by of 1 per cent. It is therefore possible by the use 
of these tables to obtain results of still greater accuracy. 
Thus, in the preceding example, the bond costs nearest to 103.35 
are 

103.05, corresponding to a yield of 4.75 per cent. 

103.68, corresponding to a yield of 4.70 per cent. 

Hence, if 103.35 corresponds to a yield of x per cent, then, 

0.33 : 0.63 =x —4.70 : 0.05. 


Hence, 


x =4.7262. 



MISCELLANEOUS PROBLEMS 


61 


MISCELLANEOUS PROBLEMS 

1. Three $1000 bonds, paying respectively 4 per cent, 5 per cent, and 

6 per cent semiannually, and all maturing in 15 years, are bought at prices 
to yield 7 per cent. Compare the prices and show that they form an arith¬ 
metical progression. 

Arts. $724.12; $816.08; $908.04. 

2. Prove that the prices of any three bonds of like denomination are in 
arithmetical progression, provided they have the same yield rate and date 
of maturity, and bear interest rates that are in arithmetical progression. 

3. From the results of Problem 2, find the value of an 8 per cent bond, 
assuming that a 6 per cent bond of the same denomination, date of maturity, 
and yield rate, is worth $98.75, while a similar 7 per cent bond is worth 
$101.25. 

4. A corporation issues $100,000 in 6 per cent bonds, redeemable at 
par in 15 years, interest payable semiannually. They accumulate a sink¬ 
ing fund at 4 pe • cent, converted semiannually. Find the semiannual 
payment necessary to meet both interest and sinking-fund charges. Con- 
side ing this semiannual payment as an annuity, and assuming that the 
bonds sold at 98, find the approximate rate of interest, that the corporation 
has to pay for the use of this money. 

5. Suppose, in Problem 4 that 20-year bonds had been issued paying 

7 per cent, semiannually, and that the issue on this basis would sell at 
102; with the same provision for a sinking fund, which plan would be more 
advantageous to the corporation? 

6. A corporation issues 5-year bonds, bearing 6 per cent interest pay¬ 
able semiannually. If they sell at 96.50, what rate of income do they 
yield? 

7. A $10,000 bond, due July 1, 1930, is sold on April 15, 1923, at a price 
to yield the purchaser 7 per cent. The bond bears 6 per cent, interest 
payable January 1 and July 1. Find its theoretical value, also its selling 
price, based on its value January 1, 1923, plus accrued interest. 

8. Find the rate of income realized on a 7 per cent bond purchased for 
102.50, paying interest semiannually, and bought 20 years before maturity 

9. An 8 per cent bond, redeemable at 103, interest payable quarterly, 
is bought at 98, 5 years before maturity. Find the rate of interest realized 
by the purchaser. 


CHAPTER V 


PROBABILITY 


41. Definition of probability. —In the toss of a coin, the 

chances of its coming “ heads” are as are also the chances of 
its coming tails;” in the throw of a die, the chances of a par¬ 
ticular face coming upward are J. With the coin, there are 
two possibilities, each equally likely, and, in the long run, half 
of the throws may be expected to be “heads” and the other 
half “ tails.” With the die, one would expect that, of a great 
number of throws, one-sixth would bring a particular face 
upward. 

This ratio of the number of favorable ways in which an 
event may happen to the total number of ways,, favorable and 
unfavorable, is used as the mathematical measure of probability. 
The definition may be algebraically stated as follows: 

If an event can happen in m ways, r of which are favorable and 

T 

s unfavorable, then the probability of a favorable occurrence is —, 

m 


and the probability of an unfavorable occurrence is —, where 


m 


r+s =m. 

It follows that, 


—= 1 ; 
m m 


T S 

hence, if p=—, then 1— p=—; therefore, if the probability 

of an event happening is p, then the probability of its not hap¬ 
pening is 1 — p. 

Again, if an event is certain, it will happen without fail in 

62 


DEFINITION OF PROBABILITY 


63 


every case; therefore 1 is the mathematical measure of cer¬ 
tainty. - 

To illustrate further: If a bag contains 7 balls, 3 of which 
are white and 4 black, then there are 7 ways of drawing a ball 
from the bag, in 3 of which the ball is white, while in 4 it is 
black. The probability of drawing a white ball is then f, and 
that of drawing a black ball y. 

While it is possible in many cases, as in the foregoing illus¬ 
tration, to enumerate accurately the total number of possible 
ways in which an event may occur favorably or unfavorably, 
there are other types in which probable future occurrences 
must be based on statistical data obtained through experience. 
Thus, if it has been observed that, of a certain class of buildings 
under similar conditions, one out of every n has been lost through 


fire each year, then - may be taken as the probability that a 

n 

particular building of this type will burn in any given year. 

The American Experience Table of Mortality (Table X), 
shows that, of 100,000 males aged 10, there will be 57,917 living 
at age 60. Hence, the probability that a particular individual 
of the original group will attain the age of 60 is 


57,917 

100,000 


0.57917. 


It should be noted, throughout this discussion, that the 
occurrence of an event, and the occurrence of any other event 
with which the given event is compared, are assumed to be 
“equally likely.” Thus, in the toss of a coin, or the throwing 
of a die, any particular face is as likely to come up as any other. 
On the other hand, in the problem just considered, it would be 
erroneous to assume that dying before the age of 60, or surviving 
that age, are equally likely events, for, if they were, the prob¬ 
ability of each would be Put another way, we assume that 
in the long run, or with a large number of cases, or trials, the 
events under consideration will occur equally often. 



64 


PROBABILITY 


Before applying the theory of probability to questions of 
insurance, it is necessary to develop some fundamental theorems. 

42. Theorems on arrangement and combination.—(i) If 
an act can be performed in a ways and, after it has been com¬ 
pleted in one of these ways, another act can be performed in b 
ways, then the two acts can be performed, in the order stated, 
in ab ways. 

The truth of this statement will be obvious, without formal 
proof. As an illustration, suppose that there are 3 routes from 
a city A to a city B, and that there are 5 ways of traveling from 
B to a third city C; then it is clear that one can go from A to C 
in 15 ways, two routes being considered different if they differ in 
any part of the journey. 

(ii) Permutations. —If, from a group of n things, sets of r are 
to be chosen and arranged in order, each arrangement is called a 
permutation. Two permutations are said to be different if 
they do not contain the same r elements throughout, or if the 
same r elements appear in both, but occur in a different order. 
The arrangement abed differs from the arrangement abce, and, 
with the same letters, the permutation abc is different from acb. 

Indicating the n elements, or objects, by the letters cq, 
a 2 , . . . a n , we may choose the first one in n ways. After one 
letter has been selected, the second one can be chosen in n — 1 
ways. Hence, by (i), the first two places can be filled in n(n — 1 ) 
ways. Proceeding in this manner, we find that the first three 
places can be filled in n(n — l)(n—2) ways, etc. Denoting 
the total number of ways of filling the r places by n P T , we have 

n Pr = n(n 1 ) (n 2 ) .... (n-r+ 1 ). . . . ( 1 ) 

In particular, if r=n, 

n P n =n(n-l)(n-2) . . . 3-2*1 =n\ ... (2) 

Thus, 

5 P 3 =5-4-3 =60, 

while 

5P5 = 5 • 4 • 3 • 2 • 1 = 5! = 120 . 


THEOREMS ON ARRANGEMENT AND COMBINATION 65 


(iii) Combinations. —When r things are to be chosen from a 
group of n things, without regard to the order of arrangement in 
any selection, the group of r objects chosen is called a combina¬ 
tion of the n things into a set of r. The total number of com¬ 
binations into sets, each containing r objects, that it is possible 
to make from the set of n, is indicated by the symbol n C r . 
Two combinations are regarded as different if one of them 
possesses an element not found in the other. 

To obtain the value of n C r , one needs only to note that, by 
permuting all of the r elements in a particular combination, r! 
permutations may be obtained. Doing this in every one of 
the n Cr combinations, we obtain a total of n C r • r! permutations. 
But every possible permutation is present in this total, since 
each permutation is present in some combination; hence, 


c -rf = P 

n \y r / . n L 


or 


nC r = 


7i^ rj 


uPr 

r\ 


(3) 


Corollary. —Every time a choice of r things is made, n—r 
things are left behind; hence, 

nC r — n C n —r . (4) 

(iv). The expansion for (a-\-b) n is given by the formula, 
(a+6) n =a n -\- n Cia n ~ l b 

-\-nC 20 ' n ~b 2 -\- . . . -\- n C r a n r b r .. . + 5 ”, . . ( 5 ) 

for n a positive integer. 

The truth of this theorem may be seen by considering a + b 
multiplied by itself n times. In the continued product, one 
forms every possible term by taking one letter from each of the 
n factors. Thus, the term a n ~ r b T can be formed in as many 
ways as the letter b can be selected r times from the n factors, 
the letter abeing chosen from the remaining n-r factors. The 
total number of such terms in the product will therefore be n C r . 





66 


PROBABILITY 


PROBLEMS 

1. Write down the values of the following symbols: 4 C 3 , 0 ^ 2 , nPi, nPi- 

2. In how many ways can 5 people be arranged in a row? 

3. How many matches of tennis singles will have to be played by 10 
players if each one is to play against every other player? 

4. Prove algebraically that n C r = nCn-r- 

5. In how many ways can 50 cards be chosen from a pack of 52? 

G. Eight people are arranged at random in a circle. What is the 
probability that a particular pair will be together? 

7. From a bag containing 5 white balls and 6 black balls, 2 are drawn 
at random. What is the probability that both will be white? Both 
black? One white and one black? What is the sum of the three results? 

8. Expand (a+6) 6 using formula (5). 

9. Five coins are tossed; what is the probability that exactly three of 
them are “heads”? 

43. Mutually exclusive events. —Two or more events are 
said to be mutually exclusive when the occurrence of any one of 
them precludes the possibility of any of the others. 

As an illustration, suppose that a single ball is to be drawn 
from a bag containing balls of different colors. The drawing 
of a ball of one color precludes the possibility of drawing one of 
another color. Thus, if there are white and red balls in the bag, 
the drawing of a white ball and the drawing of a red ball are 
mutually exclusive events. 

In the foregoing illustration, suppose that there are, in all 
13 balls in the bag, 2 of which are white and 5 red. Then the 
probability of drawing a white ball is yy, and of drawing a 
red ball is T \. The number of balls that are either red or white 
is 7. Hence the probability of drawing either a red or a white 
ball is t 7 3 , or the sum of the probability of drawing a red ball, 
and of the probability of drawing a white ball. 

From the definition of mutually exclusive events, it follows, 
then, that the probability of one or the other of them happening is 
the sum of their separate probabilities. 


COMPOUND EVENTS 


67 


44. Compound events. —Events are said to be independent 
when the occurrence of any one of them does not affect, in any 
way, the occurrence of any of the others. Thus, if two coins 
be tossed, the result for one of them is independent of the 
throw of the other. The survival of one person for a given 
number of years does not affect the probability that a second 
person will also be alive at the end of that period. These 
would be called independent events. 

When two coins are tossed, 4 different results may occur. 
Of these, for example, one result will be two heads. The prob¬ 
ability, therefore, of throwing two heads with two coins is 
|, which is the product of the probabilities that each will register 
heads. 

Again, if a coin and a die be tossed, the probability that 
a head and an ace will come up is T V, because there are 
12 possible results, only one of which consists of a head and 
an ace. But the probability that a head will come is |, and 
that an ace will be thrown is J. The probability, then that both 
of these independent events will occur is the product of their 
separate probabilities. 

The truth of this principle will be established for the joint 
occurrence of any two independent events. 

Suppose that the first can happen in a\ ways and fail in 6i 
ways, while the second can happen in 02 ways and fail in 62 
ways. Each of the 01+61 possible results of the first event 
can be associated with each of the 02+62 results of the second, 
making a total of 

(01 + 61) (02+62) =0102+0162+0261+6162 

possible cases of joint occurrence. In 0102 of these, both events 
occur. Hence, the probability that both events will happen is 

O1O2 Oi 02 

■- — ■ ■ ■ ■ ~ ■ • — 

(01 +61) (02 + 62) O1+61 O2 + 62 

Similarly, if p 1, p2, . . ., p n are the respective probabilities 
that n independent events will happen, then the probability 





68 


PROBABILITY 


that they will all occur is P1P2P2, • • • Pn', while the probability 
that they will all fail is 


(1-Pl)(l-P2) . . . (1 -Pn). 


PROBLEMS 

1. What is the chance of throwing a 1 or a 6, in a single throw of a 
die? 

2. What is the probability of throwing a 1 followed by a 6, in 2 throws 
of a die? 

3. What is the probability of throwing 3 heads in 3 throws of a coin? 

4. Find the probability of not throwing a 6 in 3 throws of a coin. 

5. If the probability of A living a certain time is , and that of B sur¬ 
viving for the same period is what is the probability that A will die and B 
will survive? That both will die? 

45. Probability with repeated trials.—If the probability of 

an event happening in one trial is p, the probability of its failing 
is q = 1 — p. Hence, the probability of its happening r times in 
n trials and failing n—r times, in any specified order, is, from 
§ 44 , equal to pq~ r . But the number of ways one can desig¬ 
nate a particular order in which the event is to happen and fail 
is the number of ways one can choose r numbers from a set of 
n, or n Cr, and these ways are all equally probable and mutually 
exclusive. Hence, the probability of the event happening 
exactly r times in n trials is n C r p r q~ r . 

Example.— Suppose the probability of an event happening in one trial 
is Find the probability that it will happen 3 times in 5 trials. There 
are 5 C 3 = 10 ways one can designate the 3 trials from among the 5 in which 
the event is to occur, and in the other two to fail. Thus, it might happen 
in the first, third and fourth, and fail in the second and fifth. The probabil¬ 
ity that the results shall be in this specified order is 

• 

6 6 6 6 0 G» \6/ ' 

But there are 10 such orders possible, in each of which the event would 



REPEATED TRIALS 


69 


happen exactly 3 times. 
3 times in 5 trials is 


Hence, the probability of the event happening 



125 

3888' 


The probability of an event happening at least r times in n trials is the 
sum of the probabilities of its happening exactly r, r + 1 , r-\-2, ... up to n 
times, or 

nC,p r q n -'+nC r + lp r+ V- r -'+ ■ ■ ■ +p"- 

In the present example, the probability of the event happening at 
least 3 times in 5 trials is then 




1 23 

+ 6* _ G48' 


It should be noticed that, by (iv) § 42, since p+p=l, we 
have 

(q+p) n = 1 =q n +nCipq n ~ 1 -\-nC 2 p 2 q n ~ 2 +. . .+nC r p r q n ~ r +. . .+p n . 

The truth of this becomes obvious if one notes that the right side 
is the sum of the probabilities that the event will fail every 
time, happen exactly once, or twice, etc., up to and including 
the probability that it will happen every time. One of these 
must happen; hence, the sum of their probabilities is 1, the 
mathematical measure of certainty. The most probable number 
of successes and failures would be given by the greatest term in 
this expansion. 

PROBLEMS 


1. Find the probability of throwing exactly 2 heads in 6 throws of a 

coin. At least 2 heads. Ans. 

2. If the chances of a team winning a particular game are f, what is the 

probability that it will win exactly 2 games out of a series of 3? At least 
2 games? Ans. yVsJ t¥s • 

3 . In a World’s Series baseball contest, the teams were rated even. 

What was the probability that one team would win 4 out of the first 5 
games played and lose the other one. Ans. 3 ^-. 

4. If, on a certain coast, one steamer is lost out of every 250 trips under¬ 
taken, what is the probability that of 10 expected to arrive at least one will 
be lost? 






70 


PROBABILITY 


5. If the probability of an event happening in one trial is what is the 
most probable number of favorable occurrences in 6 trials? 

Ans. 2. 

46. Mathematical expectation.—If p is the probability of 
obtaining a sum of money, M, then pM is the value of the 
expectation. If in a large number, m, of cases or trials the 
sum M is received a times, the average amount received for 

each trial is But, p=—; therefore, the expectation may 

m m 

be considered as the average amount received in the long run 
for each trial. 

If a person holds one ticket in a lottery containing 100 
tickets and having one prize of $25, then, in the long run, he 
may expect to win the prize once out of every 100 drawings. 
He would, therefore, pay $0.25 for each chance, or y^- of the 
prize. His expectation, then, is valued at $0.25. 

As a further illustration, suppose that 1000 men, all aged 35, 
contribute to a fund, with the understanding that each sur¬ 
vivor will receive $100 at the end of 10 years. The mortality 
tables show that approximately 907 will be alive. Hence, the 
expectation of each would be valued at $90.70. The fund will 
have to contain $90,700, if each survivor is to receive $100. 
Hence, neglecting interest, each of the 1000 men will have to 
contribute $90.70. 

47. Mortality tables.—Through the experience of insurance 
companies and other agencies, tables have been constructed 
that indicate the number of people, from a given initial group, 
that die in each succeeding year. These data are usually ex¬ 
pressed in terms of some convenient initial number, as in 
Table X, where the number of persons living at age 10 is taken 
as 100,000. 

The number of persons living at the age x is denoted by 
lx, and the number dying in the age interval from x to x+1 
by d x . The table shows the values of l x and of d x for every year 
from 2 = 10 to £=95. Three are assumed to be alive at the 
latter age, but to die during the year. The symbol (s) is used 



JOINT LIFE PROBABILITIES 


71 


to indicate a person aged x. The probability that (x) will live 
at least one year is denoted by p x , and that he will die within the 
year by q x . The following relations exist between these quan¬ 
tities : 

dx lx ^z+i*.(6) 


Vx = 


l 


£+1 


d x lx 1x4-1 

Qx = _ t~ — 7 = 1 Rx* 


(7) 


( 8 ) 


From the nature of these quantities, it is clear that 

l x =dx+d x +i+dx +2 ... to end of table. . . (9) 

lx — lx4-n =d ar +dx+l+dx+2~}“ • . . Tdx+n-1- • • (10) 


The probability that (x) will live at least n years is denoted 
by n p x ; hence, 

L 


liPx — 


l x4- n 


l 


( 11 ) 


The probability that (x) will not survive n years is denoted 
by | n q x . It follows that 

lx lx4-n 


%qx 1 nPx 


l 


. ( 12 ) 


48. Joint life probabilities.—The survival of (x) and the 

survival of (y) are independent events. If n p xy denote the 
probability that both will live n years, we have 

nPxy~nPx’nPy .(Id) 

The probability that (x) will live n years, and (y) will die 
within that period is ^ 

nPx’\nqy = nPxi\ nPy ).(14) 

The probability that at least one of two lives (x) and (y) 
will survive n years is the sum of the probabilities that both 
will live, and that either (re) or (y) will survive and the other die. 









72 


PROBABILITY 


Example 1.—Find the probability that a man aged 25 will be alive 40 
years later: 

By formula (11) 

l&b 

40P25 

fob 


From Table X, 
hence, 


2 65 = 49,341 and l 2& = 89,032; 


49,341 

i0 v 2b = — -= 0.55419. 

89,032 


Example 2.—Find the probability that A, aged 30, will live 10 years 
and B, aged 25, will die within that period. 

By formula (14), the desired probability is given by 


From Table X, 


loP 3 o(l I0P25) • 


10P30 — 


40 


30 


78,106 

85,441 


0.91415, 


10P25 — 


35 


25 


81,822 

89,032 


0.91902, 


Hence, 


I- 10 P 25 = 0.08098. 
ioP 3 o(l 10 P 25 ) =0.07403. 


PROBLEMS 

1. Find the probability that a man aged 45 will live to be 65. 

2. Find the probability that a man aged 70 will die within one year. 
Within 10 years. 

3. Calculate the probability that A, aged 35, and B, aged 40, will both 
survive 10 years. What is the probability that at least one of them will 
survive? 

4. What is the probability that three persons, each aged 21, will all 
reach the age of 60? What is the probability that none will reach that 
age? 

5. What is the probability that three persons, A, B, and C, all of the 
same age, will die in any given order, such as A, B, C? 

6. A boy aged 15 is to receive $10,000 on his twenty-first birthday. 
What is the value of his expectation on a 5 per cent basis? 





PROBLEMS 


73 


7. Prove that n p x = p x • y x +i • Pi+ 2 - . .. p*+n-i. 

8 . What is the probability that a man aged 21 will live 40 years? If he 
is alive after 20 years, what is the probability that he will then live 20 years 
more? Compare the two answers. 

9. If one house out of 800, of a certain class, is totally destroyed by 
fire each year, what is the value of the expectation of a man who insures 
such a house for $10,000? What should be the net annual cost of such an 
insurance policy? 

10. Using the data of Table X, plot the curve showing the probability 
of dying for each age. When is it a minimum? 


CHAPTER VI 


LIFE ANNUITIES 


49. Definition of life annuity.—A life annuity in its simplest 
form, is one whose payments continue only during the lifetime 
of the individual concerned. Its present value, or cost, will 
depend not only upon the rate of interest, but also upon the 
probability of living. In this latter respect it differs from 
annuities certain, considered in Chapter II; in that case the 
term, or number of payments, was assumed to be definite. 

The present value of a life annuity will be obtained, for any 
assumed rate of interest, in terms of the probability of survival 
to any given age. This is accomplished by computing the 
present value of the expectation at each succeeding age and 
adding the results. 

50. Pure endowment.—The present value of the expecta¬ 
tion of a person aged x, who is to receive 1 if he lives to the age 
x-f -n is called an n-year pure endowment of 1. The present 
value of 1 due in n years is v n , and the probability of a person 
aged x living n years is n p x . Denoting the present value of a 
pure endowment of 1 by n E x , then by the definition 


n E x =v n - n Px = v n ■ 


vx-\~ n 


■ • ( 1 ) 


which is the present value of the expectation, as defined in §46. 


Example. —Find the value on a 5 per cent basis, of a pure endowment of 
$1000, payable 10 years hence, to a person now aged 25. 

The present value is given by 


where, 


IOOO10E25 — 1000 y 10 * 1 o 7 ? 25 j 


10P25 


hs 81,822 
Z 25 $9,032* 
74 




COMPUTATION OF LIFE ANNUITY 


75 


Upon performing the indicated operations, it is found that 


$1000- 10 E 2 5 = $564.20. 


PROBLEMS 

1. An estate valued at $25,000 is left to an heir aged 15, to be given him 
upon his attaining the age of 21. Find the present value of the inheritance 
upon a 6 per cent basis. 

2. Find the present value of a pure endowment of $1000, to a person 
aged 25, computed upon a 5 per cent basis, (i) payable if he attains the age 
of 50, (ii) payable if he survives age 75. 

3. An heir, aged 12, is to receive $50,000 when he attains the age of 21. 
What is the present value of his expectation upon a 5 per cent basis? 

4. Two brothers, one 15, the other 18, are each to receive $10,000 upon 
reaching the age of 21. Find the present value of the expectation of each 
on a 5 per cent basis. 

5. Find the values of the expectations in Problem 4, one year later. 

51. Computation of life annuity.—From the definition, the 
present value of a life annuity of 1, to a person aged x, consists 
of the sum of pure endowments of 1, for each succeeding age. 
Denoting by a x the present value, or cost of such an annuity, 
we have 

«x = i^x+ 2 -^x+ 3 ^x+ ... to table limit, ... (2) 

_vl x +i-\-v 2 l x -t2-i-v 3 lx+3-\- ... to table limit 
— ? . . . (o) 

Lx 

This result may also be obtained, without introducing the 
notion of probability, by supposing that each of a group of l x 
people, all aged x , purchase such an annuity. The total 
amount that must be contributed by the group must be such 
that each of the survivors, at each succeeding age, may receive 1. 
Thus, at age x+1 there will be l X/:t i persons alive; hence, an 
amount vl x+ 1 must be set aside now to provide a sum l x+ 1 at 
the end of the first year. Similarly, an amount l x+ 2 will have 
to be paid at the end of the second year, and its present value 
is v 2 l x+ 2 . Proceeding in this manner for each succeeding year, 
until all of the original l x persons are dead, we have, as the total 



76 


LIFE ANNUITIES 


amount that must be set aside now to meet the future pay¬ 
ments, the sum 

vl x +i+v 2 l x+ 2 +v 3 l x+ 3 + ... to table limit. 

This amount, however, is to be shared equally by all of the 
original l x persons. Hence, each must contribute 

vl x +\-\-V 2 lx+2-\-vH x +3-\- • • • 

7 

L X 

which is identical with a x , as given by Formula (3). 

The value of an endowment of R instead of 1, is clearly 
R n E x , and the present value of a life annuity of R is Ra x . 

The labor of computing a x in the form given by (3) would be 
very great. This will be lessened, however, by the transforma¬ 
tions shown in the next article; but it should be noted that the 
value of a life annuity at age x can be obtained from the value 
at age :r+l by a simple calculation. The present value of 
a x+ 1 is va x+ i. Hence, the total present cost of l x+ 1 life annu- 
ties for the l x+ 1 survivors at age £+1, together with the pay¬ 
ment of 1 to each at the end of the first year, is 

1 “b VQ>x -\-1 ’ ^-c+1 • 

Hence, each of the l x original persons will have to pay an 
amount. 

a x = v(l+a x+ i)~± .(4) 

L x 

Thus, if the a x+ 1 be known, the value of a x can be easily found, 
and indeed, by starting with the most advanced age in the mor¬ 
tality table, a complete table of life annuities could be con¬ 
structed, for each rate of interest desired. 

PROBLEMS 

1 . From the American Experience Table of Mortality, using per 
cent interest, the value of a life annuity of 1 at age 35 is 17.614. Find the 
value at age 34. How many years, approximately, would an annuity 
certain have to run to cost the same amount? 




COMMUTATION COLUMNS 


77 


2. Given, a 5 o = 13.535; find, by successive steps, a 49 , a 48 and a 47 . Inter¬ 
est at 3^ per cent. 

3. Given, a 20 = 20.144; find the value of a 2 i. Interest is at 3^ per cent. 

4. By using Formula (3) and the mortality table, find a 90 on a 3£ 
per cent basis. Compute it also on a 5 per cent basis. 

52. Commutation columns.—While life annuities may be 
calculated by the preceding methods, the work is greatly facil¬ 
itated by certain tables known as commutation columns. 

Starting with a x , as given by formula (3), we have 

vl x+ i-\-v 2 l x+ 2 ~\-v 3 l x+ 3 -\- ... to table limit 
u x — 1 , . 


Multiplying numerator and denominator by v x , it becomes 


if +1 l z+i -\-v x+2 l x + 2 -\-v x+s lx+ 3 ~\~ ... to table limit 

v% 



If, now, we denote the product v x+k i x+1c by the symbol 
D x+k , we have 


a x 


Dx+i -\-D x+ 2~\-D x +3-{- 

~D X 


. . to table limit 



Placing the numerator equal to N x +1 (5) becomes 


where 


a x 


N x+l 

D x 


N x+ 1 =D x+ i+D x+2 -\-D x+ 3 + ... to table limit. 


( 6 ) 

(7) 


The quantities N x and D x are known as commutation sym¬ 
bols, and their values for each age are given in Table XI. 

53. Deferred annuities.—If the payments of a life annuity 
are to begin at the end of n-\-l years, instead of at the end of one 
year, the annuity is said to be deferred n years. 

The cost, or present value, of such an annuity is then less 
than that of a life annuity whose payments begin the first 
year by the cost of the endowments for the first n years. Denot- 








78 


LIFE ANNUITIES 


ing the present value of the life annuity of 1 deferred n years, 
for a person aged x, by n \a x , we have 

n\(lx =n+lEx-\-n+2E x -\- n +zE x -\- ... to table limit. . (8) 

Replacing each one of the E ’s by its value, given by (1), (8) 
becomes 

, v n+1 l x+ n+i-\-v n+2 l x+n + 2 + . . • to table limit , , 

rvflx t • • \*v 

lx 

As in § 52, this may be expressed in terms of D’ s, by multi¬ 
plying numerator and denominator by if. Whence, 


D x+n+ i+Z) I+n+2 + . • . to table limit 

IK " 

Nx-\- n+1 

~d*~~ . 


( 10 ) 

(ii) 


54. Temporary annuities.—When the payments of an 
annuity are to cease at the end of n years, provided the annui¬ 
tant lives that long, it is called a temporary annuity for n years. 
It provides for payments during the life of the insured up to 
the end of n years, but no longer. 

The present value of a temporary annuity of 1, for a person 
of age x, to run n years, is denoted by a x ^. It is clear that 
the cost of a temporary annuity for n years, and of one deferred 
n years, together make up a whole life annuity. Hence, 

Ux;i| = dx n|^X‘ 

_ N x +i —AT+n+i n o\ 

D x . 

Table XI gives the values of N x and of D x on the basis of the 
American Experience Table of Mortality, interest being allowed 
at 3J per cent. Unless otherwise stated, all computations are 
assumed to be made on this basis. 








ANNUITIES DUE 


79 


PROBLEMS 

1. Using Formula (6) and Table XI, find the value of a 2 o. 

2. Find the present value of a life annuity to a person aged 50, the 

annual payment to be $1000. Ans. $13,534.72. 

3. Find the cost of the annuity in Problem 2, taken out at the age of 

50, the payments being deferred 10 years. Ans. $5901.04. 

4. Find the cost of a temporary annuity of $1000 per year, taken out 

at the age of 50, and terminating in 10 years. Ans. $7633.68. 

5. What relation exists between the results of Problems 2, 3 and 4? 
Explain the reason. 

6 . Find the present value of a life annuity of $1000 per year, bought by 
a man aged 40, payments to begin 20 years later. 

7. Compute 20 !a 30 and a 30 ^|. 

55. Annuities due.—If the first payment is made at once 
instead of at the end of the year, and each succeeding payment 
is also made in advance, the annuity is called an annuity due. 
The present value of a life annuity due of 1, payable annually 
to a person aged x, is denoted by a x (cf. § 19). 

An annuity due differs from an ordinary annuity only by 
the additional first payment. Hence, 


But, from equation (6), 


a x — 1T u 3 


(13) 


a x = 


N x+ 1 . 


hence, 


a x = l-f 


D x ’ 

N x +1 D x -\-N x +i N a 




D x 


d; 


• • 


. (14) 


Similarly, the present value of a deferred annuity due, 
the first payment to be made at the beginning of the nth year, 
is the same as the present value of an ordinary annuity deferred 
n -1 years. Hence, using the corresponding notation, 




N X -\- n 



(15) 








80 


LIFE ANNUITIES 


A temporary annuity due, to run n years, is the difference 
between a whole life annuity due and an n-year deferred life 
annuity due. Hence, 


&XTl\ tl\&X 


N x - N x+: 

D x 


( 16 ) 


MISCELLANEOUS PROBLEMS 

1. Compute a 30 , a 3 „, i 0 |a 30 and 10 |a 80 . 

2. A will provides that an heir, aged 35, is to receive $1000 per year, 
payable in advance, for life, what is the present value of the expected 
payments? 

3. If an estate of $25,000 is to be turned into cash, and paid in the form 
of a life annuity to an heir, aged 50, what would be the annual payment? 

4. What would the annual payment be in Problem 3, if the annuity 
were payable for 20 years, contingent upon the survival of the beneficiary? 

5. Under the first pension plan of the Carnegie Foundation, the retire¬ 
ment age was 65. This was later changed to 70 for the corresponding 
pension. Compare the present values of the expectations of a professor 
aged 35 under the two plans, assuming that his retiring allowance in either 
case would be $3000, and payable at the end of each year. 

6 . A person aged 50 purchases a life annuity for $25,000. What is the 
annual income? 

7. If, in Problem 6, the payments are to be deferred 10 years, what 
will be the annual income? 

8 . How much would a 10-year temporary life annuity yield if purchased 
at age 55 for $10,000? 

9. A person aged 60, deeded to a university, property valued at $20,000, 
in consideration of an equivalent life annuity on a 3| per cent basis. Find 
the annual income. 

10. A person aged 21 receives an inheritance of $3000 per year for life, 
payable at the beginning of each year. An inheritance tax of 4 per cent is 
to be paid on its present value Find the amount of the tax. 



CHAPTER VII 


ELEMENTARY PRINCIPLES OF LIFE INSURANCE 

56. Introduction.—The mathematical treatment of life 
insurance involves many complex problems. It will be the 
purpose of this chapter, however, to consider only certain basic 
principles upon which the theory rests, and to apply these 
principles to the determination of the cost of some typical 
forms of insurance. 

The fundamental principle of all insurance is that a large 
group of persons contribute, through the agency of the insur¬ 
ance companies, for the losses sustained by members of the 
group. The money paid by the insured for his protection is 
called the premium, and the contract between him and the 
company is called the policy . It is necessary to determine 
beforehand the amount of the premium sufficient to provide 
for the probable losses. Indeed, premiums are always collected 
in advance, usually in annual, or semiannual installments, 
sometimes even more frequently. It becomes important, 
therefore, to determine the proper amount to be paid by the 
policyholder for the particular form of policy desired. 

The principal forms of life insurance fall into two classes— 
whole-life and term insurance. In whole-life insurance, the 
company contracts to pay a certain sum upon the policy¬ 
holder’s death. With term insurance, payment is made only if 
the insured dies within a specified number of years. 

The two most important elements that enter into the deter¬ 
mination of the premium are the death rate, and the rate of 
interest that can be realized on investments. When only these 
two are taken into account, the cost thus determined is called 
the net premium. After this has been determined the company 

81 


82 ELEMENTARY PRINCIPLES OF LIFE INSURANCE 


must increase it sufficiently to cover expenses of all kinds. 
This last process is called loading, and the final amount charged 
for the insurance is called the gross, or office, premium. The 
present discussion is concerned only with the determination of 
the net premium. 

The mathemcatical determination of the net premium will 
be based on the assumptions that deaths will occur with exactly 
the frequency indicated in the mortality tables, and that earn¬ 
ings will be exactly those resulting from the assumed rate of 
interest. Furthermore, benefits resulting from deaths in any 
particular year will be regarded as paid at the end of the year. 
In this connection, year means the policy year. Twelve calen¬ 
dar months from the date of issue of the policy is the first policy 
year, the next twelve months is the second policy year, and 
so on. 

In all problems connected with life insurance, the results 
will be derived, as in the case of life annuities, on the basis of 
a benefit of 1. The solution of the questions that arise are 
similar to those connected with life annuities, but the probability 
of dying, rather than that of living, is now under considera¬ 
tion. 

57. Net single premium. Whole-life policy.—The present 
value, or net cost, of a whole-life insurance policy, expressed as 
a single sum, is known as the net single premium. 

Using the mortality tables, and assuming a fixed rate of 
interest, suppose that l x people, all of age x, pay to an insurance 
company a sum sufficient to pay 1 for each death as it occurs. 
The net single premium that each of the l x persons will have to 
pay will be the total present value of all future death claims, 
divided by l x . 

At the end of the first year, the company will be called on to 
pay an amount d x ; therefore they must now have on hand a 
sum vd x , which, with interest, will take care of these claims 
at the end of the year. Similarly an amount v 2 d x+l will have 
to be provided now to take care of death claims at the end of 
the second year, and so on until all of the original l x persons 


COMMUTATION SYMBOLS 


83 


have died. The sum of the present values of all these pay¬ 
ments is, 

vd x J rv 2 d x +i-\-v i d x + 2 -\- ... to table limit. . . (1) 

But this cost is to be shared equally by all of the original l x 
persons buying insurance. Hence, each would pay an amount 
which will be denoted by A x , given by the formula, 

A _vd x -\-v 2 d x+ i-\-v 3 d x+ 2 -{- ... to table limit , . 

lx 

58. Commutation symbols.—Formula (2) could be used 
for the determination of A x , but the calculations may be facil¬ 
itated by the following transformation, analogous to that used 
in § 52. 

Multiplying numerator and denominator by v x , the denom¬ 
inator becomes D x , and the terms of the numerator are of the 
form 

v x+k+1 -d x+t . 


Defining C x by the equation, 


C x =v x+l d. 


Xj 


(3) 


we have 




C X +Cx+i + CX+ 2 + .... to table limit 


D x 


. . . (4) 


Defining M x by the equation, 

M x =C x +C x +i+C x + 2 + ... to table limit, . . (5) 


we finally obtain 


A x 


M* 

Di 


• 4 ' • 


( 6 ) 


The values of D x and M x , interest at per cent, are found 
in Table XI. 






84 ELEMENTARY PRINCIPLES OF LIFE INSURANCE 


PROBLEMS 


1 . Find the net single premium on a whole-life policy of $1000, on the 
life of a person aged 30. 

2. Find the net single premium for a policy of $1000 on the life of a 
person aged 70. 

3 . Compare the costs of whole-life policies of $10,000 for ages 20 and 21, 
and for ages 70 and 71, noting at which period of life the annual change in 
cost is the greater. 


59. Annual premiums.—It is customary to pay insurance 
premiums in equal annual installments. They may also be paid 
semiannually, or even more frequently. Again, they may con¬ 
tinue throughout the life of the insured, or they may run for a 
limited period, say n years, and then cease, even though the 
insurance continues in the form of a whole-life policy. If the 
payments continue throughout the life of the insured, the 
policy is called an ordinary life 'policy. If the premiums are to 
cease after n years, the policy is called an n-payment life policy. 

In either of the foregoing cases, the net annual premium 
is that sum which, if paid at the beginning of each policy year, 
is equivalent to the net single premium. The annual premiums, 
therefore, constitute an annuity due, whose present value is A x . 

If P x denotes the net annual premium for an ordinary life 
policy, purchased at age x, for an insurance of 1, then, by 
§ 55, formula (13), 


But by (6) 


hence, 


A x — P x o. x — P x (l+a x ) =P X ~^. 

■L'x 

A =— 

^ * D x ’ 

P 

* Z —' 


• ( 7 ) 


N , 


• (8) 


The net annual premiums for an n-payment life policy con¬ 
stitute a temporary annuity for n-1 years, together with the 
initial payment at the beginning of the first policy year. Denot- 





NET SINGLE PREMIUM FOR TERM INSURANCE 85 


ing by n P x the amount of the premium for an insurance of 1, 
we have 



or, 



(9) 


By (12), § 54, and (7) § 52, 

N x N x+ n 



X 


Hence, substituting in (9), and replacing A x by its value, as 
before, we have 



( 10 ) 


PROBLEMS 


1. Find the net annual premium for an ordinary life policy of $1000, on a 
life aged 21. 

2. Find the net annual premium for an ordinary life policy of $1000, 
on a life aged 50. 

3. Find the net annual premiums for a 20-payment life policy for $1000, 
lor ages 21 and 50 respectively. Compare your answers with those in 
Problems 1 and 2. 

60. Net single premium for term insurance.—As explained 
in § 56, a term insurance policy is a contract to pay the face of 
the policy if, and only if, death occurs within the stated term 
of years. This form of insurance is written for periods of vari¬ 
ous lengths, but usually for five years, or a multiple of five 
years. It is usually bought by a person desiring protection for 
his estate, covering some period within which he is to be engaged 
in a business enterprise that would suffer loss if he should die 
before it was sufficiently developed. 

The net single premium for term insurance of 1 for n years, 
on the life of a person aged x, will be denoted by the symbol 
n A x . Its value will be found in the same manner as used in 
§ 57 for the determination of A x . 










86 ELEMENTARY PRINCIPLES OF LIFE INSURANCE 


Suppose that l x persons, all of age x, purchase n-year term 
policies. The present value of the death claims for each of the 
n years will be respectively vd x , v 2 d x + i, .... v n d x + n -\. The 
sum of these quantities gives the total cost, to be shared equally 
by the l x persons buying the insurance. The amount, \ n A XJ 
that each will pay is therefore 



vd x +v 2 d x+ 1 + . . . +v n d n+x+ \ 

h 


( 11 ) 


If numerator and denominator be multiplied by i *, the 
symbols C x and D x may be introduced, giving 


By (5), 
and, 


A C x -\-C x +i~{-C x + 2 I • • • I C X -\- n —1 /lo\ 

x = Jy • • • V-*-"/ 


M x = C x -\-C x +\-\- ... to table limit, 
x -\- n — C x + nT C n _j_ i+ ... to table limit. 


Hence, the numerator of (12) is the difference between these 
two expressions, so that 



M x -M x+n 

D x 


(13) 


61. Net annual premium for term insurance. —The net 

annual premium, n P x , may be considered as the annual pay¬ 
ments of a temporary annuity due. Hence, 

7>Px’&xn\ ~ n,A x . .(Id) 

Substituting the value of \ n A x from (13), and the value of 
a x - { from (16) § 55, we have, 

p _M x ~M x + n 

* r x—^F nr. {ID) 


N x -N 


x-\- n 


PROBLEMS 

1. Find the net single premium for term insurance of $25,000 for 5 
years, on the life of a person aged 40. 

2. What would the net annual premium be for the policy in Problem 1? 










ENDOWMENT INSURANCE 


87 


3. Find the net single premium for term insurance of $10,000 for 10 
years on the life of a person aged 50. Compare it with the net single pre¬ 
mium for a whole life policy for the same amount. 

4. A person aged 60 buys 5-year term insurance of $100,000. Find the 
net annual premium. 

62. Endowment insurance.—An endowment insurance is 
an agreement to pay the face of the policy in the event of the 
death of the insured within a certain specified period, called the 
endowment period, and it also provides that the face of the 
policy will be paid at the end of the period, if the insured sur¬ 
vives. 

If the period be denoted by n, then endowment insurance 
consists of term insurance for n years, together with an n-year 
pure endowment. 

Denoting the net single premium for endowment insurance 
of 1 issued to a person aged x, by A x ^, we have the relation, 

Axn\ =\nA x ~{-nE X .( 16 ) 

But, by (1) § 50, 

7)^7 ft\ 

jp _ u l x+ n _ u n 

n v% * 

By definition (§ 52), 

D x = v x l x , 

hence, 

. (17) 


Introducing the value of n A x from (13), 



M x — M x+ n~f~ Dx-\- n 

D x 


(18) 


The net annual 'premium for an 7^-year endowment policy of 
1, for a person aged x, may be obtained by regarding A x „ { as 
the present value of an annuity due, to run n years. Hence, if 
its value be denoted by P^, 










88 ELEMENTARY PRINCIPLES OF LIFE INSURANCE 


hence, 



M x —M x + n -\-D x -[. n 

N x N x + n 


( 20 ) 


PROBLEMS 

1. Find the net annual premium on a 20-year endowment policy for 
$10,000, purchased at age 21. What would the premium be if term insur¬ 
ance had been purchased for the same period? 

2. Find the net annual premiums on $1000 policies, purchased by a man 
aged 45, for the following types: 

(a) 20-year endowment. 

( b ) 20-year term. 

(c) Whole life. 

3. Find the net single premium on a $10,000 endowment policy, pur¬ 
chased at age 25. What would it be if purchased at age 60? 

63. Valuation of policies. Reserves.—The probability of 

dying, except for the very young, increases from year to year. 
If, then, a person insured himself year by year, his premium 
would increase with advancing age. It follows, therefore, that 
when he takes out a whole life policy, paying a uniform or level 
premium throughout, he pays more in the earlier years, and less 
in the later years, than is required by the natural, or year by 
year, premium. 

The excess paid in the earlier years,, over mortality require¬ 
ments, is held by the company and, with its interest earnings, 
takes care of the deficiency that will occur in the later years. 
This amount is known as the terminal reserve on the policy, 
and is a liability of the insurance company to its policy¬ 
holders. 

To determine the terminal reserve at the end of any given 
year, after the policy has been issued, one notes that the present 
value, at that time, of the unpaid premiums, together with 
the terminal reserve, are equal to the net single premium of a 
policy taken out at the age then attained. If n denotes the 
number of years since the policy was issued, and n V x the 



GROSS PREMIUM. LOADING 


89 


reserve dt that time on a policy issued on a life aged x years, 
then, 

Ax+n = n ^ x~\~Px{\ ~\~dx-\-7i) .(21) 

From this equation, 


nV x — A x+n ~Px(l-\-dx + n) .( 22 ) 

d his formula may be expressed in terms of commutation 
symbols. From (6), 

M , 


Formula (8) gives 


j _^x+n 

S*-x+ n ~ • 

Ux+ n 


p — ^ n 

x N x ’ 


while (6) and (7) § 52 give 

N x + n 


1 +&X+ n — 


D 


x+ n 


Substituting these values in (22), 


T/ _ Afx+n'N x —M x -N x + n 
n x ~ • 


x+ n 



PROBLEM 

Find the terminal reserve in the fifth policy year on an ordinary life 
policy of $1000, taken out at age 21. 

64. Gross premium. Loading.—In the preceding articles 
it has been seen that the net premiums are the mathemat¬ 
ical equivalent of the benefits, based, however, on the assump¬ 
tion of a low rate of interest, usually 3J per cent. Earnings of 
the insurance company, over and above this rate, go into the 
general surplus fund, but, in addition to this, it is necessary to 
provide funds for expenses incident to the business. These 
include agent’s commissions, cost of medical examinations, the 
general expenses of administration, etc. 

These costs are met by increasing, or loading , the premium. 







90 ELEMENTARY PRINCIPLES OF LIFE INSURANCE 


This is sometimes done by adding a fixed percentage of the net 
premium, uniform for all ages; in other cases the percentage 
varies for different ages. These plans may be also combined 
with a loading by addition of a constant charge. 

The net premium, thus increased by the “loading” process, 
is called the gross, or office, premium. This is the amount 
actually paid by the policyholder. Although the methods of 
loading may differ from company to company, they usually 
result in substantial agreement for policies of the same kind. 

The gross premium thus determined is based on conserva¬ 
tive estimates of probable expenditures and claims. In cal¬ 
culating the net premium, a low rate of interest is assumed. 
The earnings of the company will, in general, be substantially 
larger than those estimated at this rate. The mortality table 
is also constructed so as to give conservative results with 
respect to probable death claims, and the loading for expenses 
incident to the business is designed to cover adequately all 
such expenditures. 

At the end of each year, then, there should be a surplus 
remaining, after all expenses have been met and funds have 
been set aside to meet all reserves, which, as already seen, 
constitute a liability against the company. In the mutual 
companies, this surplus belongs to the policyholders, and is 
returned to them in the form of dividends , or credited to the 
policy as additional insurance. 

65. Conclusion.—The purpose of the last two chapters has 
been to show what mathematical principles enter into the fun¬ 
damental problems of life insurance. These have been for¬ 
mulated in as simple a manner as possible, but the matters 
presented are merely an introduction to the subject. 

Students interested in a further study of insurance are 
referred to such books as Moir’s “Life Assurance Primer,” 
and the “Institute of Actuaries Text-Book.” 


TABLE I—THE NUMBER OF EACH DAY OF THE YEAR 


Day of 
Month 

I 

Jan. 

i 

Feb. 

Mar. 

i 

April 

May 

__ i 

June 

July 

Aug. 

H 

li. 

W 

GG 

h 

O 

O 

Nov. 

Dec. 

Day of 

Month 

1 

1 

32 

60 

91 

121 

152 

182 

213 

244 

274 

305 

335 

1 

2 

2 

33 

61 

92 

122 

153 

183 

214 

245 

275 

306 

336 

2 

3 

3 

34 

62 

93 

123 

154 

184 

215 

246 

276 

307 

337 

3 

4 

4 

35 

63 

94 

124 

155 

185 

216 

247 

277 

308 

338 

4 

5 

5 

26 

64 

95 

125 

156 

186 

217 

248 

278 

309 

339 

5 

6 

6 

37 

65 

96 

126 

157 

187 

218 

249 

279 

310 

340 

6 

7 

7 

38 

66 

• 97 

127 

158 

188 

219 

250 

280 

311 

341 

7 

8 

8 

39 

67 

98 

128 

159 

189 

220 

251 

281 

312 

342 

8 

9 

9 

40 

68 

99 

129 

160 

190 

221 

252 

282 

313 

343 

9 

10 

10 

41 

69 

100 

130 

161 

191 

222 

253 

283 

314 

344 

10 

11 

11 

42 

70 

101 

131 

162 

192 

223 

254 

284 

315 

345 

11 

12 

12 

43 

71 

102 

132 

163 

193 

224 

255 

285 

316 

346 

12 

13 

13 

44 

72 

103 

133 

164 

194 

225 

256 

286 

317 

347 

13 

14 

14 

45 

73 

104 

134 

165 

195 

226 

257 

287 

318 

348 

14 

15 

15 

46 

74 

105 

135 

166 

196 

227 

258 

288 

319 

349 

15 

16 

16 

47 

75 

106 

136 

167 

197 

228 

259 

289 

320 

350 

16 

17 

17 

48 

76 

107 

137 

168 

198 

229 

260 

290 

321 

351 

17 

18 

18 

49 

77 

108 

138 

169 

199 

230 

261 

291 

322 

352 

18 

19 

19 

50 

78 

109 

139 

170 

200 

231 

262 

292 

323 

353 

19 

20 

20 

51 

79 

110 

140 

171 

201 

232 

263 

293 

324 

354 

20 

21 

21 

52 

80 

111 

141 

172 

202 

233 

264 

294 

325 

355 

21 

22 

22 

53 

81 

112 

142 

173 

203 

234 

265 

295 

326 

356 

22 

23 

23 

54 

82 

113 

143 

174 

204 

235 

266 

296 

327 

357 

23 

24 

24 

55 

83 

114 

144 

175 

205 

236 

267 

297 

328 

358 

24 

25 

25 

56 

84 

115 

145 

176 

206 

237 

268 

298 

329 

359 

25 

26 

26 

57 

85 

116 

146 

177 

207 

238 

269 

299 

330 

360 

26 

27 

27 

58 

86 

117 

147 

178 

208 

239 

270 

300 

331 

361 

27 

28 

28 

59 

87 

118 

148 

179 

209 

240 

271 

301 

332 

362 

28 

29 

29 


88 

119 

149 

180 

210 

241 

272 

302 

333 

363 

29 

30 

30 


89 

120 

150 

181 

211 

242 

273 

303 

334 

364 

30 

31 

31 


90 


151 


212 

243 


304 


365 

31 


Note. —For leap years the number of the day is one greater than the tabular 
number after February 28. 


91 




































92 


TABLES 


TABLE II—AMOUNT OF 1 
s= (l + i) w 


n 

Vz% 

1% 

1 Vi % 

iy 2 % 

n 

1 

1.005 0000 

1.010 0000 

1.012 5000 

1.015 0000 

1 

2 

1 .010 0250 

1.020 1000 

1.025 1562 

1.030 2250 

2 

3 

1.015 07 51 

1.030 3010 

1.037 9707 

1.045 6784 

3 

4 

1.020 1505 

1.040 6040 

1.050 9453 

1.061 3636 

4 

5 

1.025 2512 

1.051 0100 

1.064 0822 

1.077 2840 

5 

G 

1.030 3775 

1.061 5202 

1.077 3832 

1.093 4433 

6 

7 

1.035 5294 

1.072 1354 

1.090 8505 

1.109 8449 

7 

8 

1.040 7070 

1.082 8567 

1 .104 4861 

1.126 4926 

8 

9 

1.045 9106 

1.093 6853 

1.118 2922 

1.143 3900 

9 

10 

1.051 1401 

1.104 6221 

1.132 2708 

1.160 5408 

10 

11 

1.056 3958 

1.115 6684 

1.146 4242 

1.177 9489 

11 

12 

1.061 6778 

1. 126 8250 

1 .160 7545 

1. 195 6182 

12 

13 

1.066 9862 

1.138 0933 

1 . 175 2640 

1.213 5524 

13 

14 

1.072 3211 

1 . 149 4742 

1. 189 9548 

1.231 7557 

14 

15 

1.077 6827 

1. 160 9690 

1.204 8292 

1.250 2321 

15 

16 

1.083 0712 

1.172 5786 

1.219 8896 

1.268 9856 

16 

17 

1.088 4865 

1.184 3044 

1.235 1382 

1.288 0203 

17 

18 

1.093 9289 

1 . 196 1475 

1.250 5774 

1.307 3406 

18 

19 

1.099 3986 

1.208 1090 

1.266 2096 

1.326 9508 

19 

20 

1.104 8956 

1.220 1900 

1.282 0372 

1.346 8550 

20 

21 

1.110 4201 

1.232 3919 

1.298 0627 

1.367 0578 

21 

22 

1.115 9722 

1.244 7159 

1.314 2885 

1.387 5637 

22 

23 

1 . 121 5520 

1.257 1630 

1.330 7171 

1 .408 3772 

23 

24 

1.127 1598 

1.269 7346 

1.347 3510 

1.429 5028 

24 

25 

1 .132 7956 

1.282 4320 

1.364 1929 

1.450 9454 

25 

26 

1.138 4596 

1.295 2563 

1.381 2454 

1.472 7095 

26 

27 

1. 144 1518 

1.308 2089 

1.398 5109 

1.494 8002 

27 

28 

1.149 8726 

1.321 2910 

1.415 9923 

1.517 2222 

28 

29 

1.155 6220 

1.334 5039 

1.433 6922 

1.539 9805 

29 

30 

1.161 4001 

1.347 8489 

1.451 6134 

1.563 0802 

30 

31 

1.167 2071 

1.361 3274 

1.469 7585 

1.586 5264 

31 

32 

1.173 0431 

1.374 9407 

1.488 1305 

1.610 3243 

32 

33 

1 .178 9083 

1.388 6901 

1.506 7321 

1.634 4792 

33 

34 

1 .184 8029 

1.402 5770 

1.525 5663 

1.658 9964 

34 

35 

1.190 7269 

1.416 6028 

1.544 6359 

1.683 8813 

35 

36 

1.196 6805 

1.430 7688 

1.563 9438 

1.709 1395 

36 

37 

1.202 6639 

1.445 0765 

1.583 4931 

1.734 7766 

37 

38 

1.208 6772 

1.459 5272 

1.603 2868 

1.760 7983 

38 

39 

1.214 7206 

1.474 1225 

1.623 3279 

1.787 2102 

39 

40 

1.220 7942 

1.488 8637 

1.643 6195 

1 .814 0184 

40 

41 

1.226 8982 

1.503 7524 

1.664 1647 

1.841 2287 

41 

42 

1.233 0327 

1.518 7899 

1.684 9668 

1.868 8471 

42 

43 

1.239 1979 

1.533 9778 

1.706 0288 

1 .896 8798 

43 

44 

1.245 3938 

1.549 3176 

1.727 3542 

1.925 3330 

44 

45 

1.251 6208 

1.564 8108 

1.748 9461 

1.954 2130 

45 

46 

1.257 8789 

1.580 4588 

1.770 8080 

1.983 5262 

46 

47 

1.264 1683 

1.596 2634 

1.792 9431 

2.013 2791 

47 

48 

1.270 4892 

1.612 2261 

1.815 3548 

2.043 4783 

48 

49 

1.276 8416 

1.628 3483 

1.838 0468 

2.074 1305 

49 

50 

1.283 2258 

1.644 6318 

1.861 0224 

2.105 2424 

50 

60 

1.348 8502 

1.816 6967 

2 107 1814 

2.443 2198 

60 

70 

1.417 8305 

2.006 7634 

2.385 9000 

2.835 4563 

70 

80 

1.490 3386 

2.216 7152 

2.701 4849 

3.290 6628 

80 

90 

1.566 5547 

2.448 6327 

3.058 8126 

3.818 9485 

90 

100 

1.646 6685 

2.704 8138 

3 .463 4043 

4.432 0456 

100 














AMOUNT OF 1 


93 


TABLE II—AMOUNT OF 1—Continued 
s = (1 + i) n 


n 

l 3 /4% 

2% 

2V 2 % 

3% 

n 

1 

1.017 5000 

1 .020 0000 

1.025 0000 

1.030 0000 

1 

O 

1.035 3062 

1.040 4000 

1 .050 6250 

1 .060 9000 

2 

3 

1.053 4241 

1.061 2080 

1 .076 8906 

1.092 7270 

3 

4 

1.071 8590 

1.082 4322 

1.103 8129 

1.125 50S8 

4 

5 

1.090 6166 

1.104 0808 

1.131 4082 

1.159 2741 

5 

6 

1.109 7024 

1.126 1624 

1.159 6934 

1.194 0523 

6 

7 

1.129 1222 

1.148 6857 

1.188 6858 

1 .229 8739 

7 

8 

1 . 148 8818 

1.171 6594 

1.218 4029 

1.266 7701 

8 

9 

1 . 168 9872 

1.195 0926 

1.248 8630 

1.304 7732 

9 

10 

1.189 4445 

1.218 9944 

1.280 0845 

1.343 9164 

10 

11 

1.210 2598 

1.243 3743 

1.312 0867 

1.384 2339 

11 

12 

1.231 4393 

1.268 2418 

1.344 8888 

1 .425 7609 

12 

13 

1.252 9895 

1.293 6066 

1 .378 5110 

1 .468 5337 

13 

14 

1.274 9168 

1.319 4788 

1.412 9738 

1.512 5897 

14 

15 

1.297 2279 

1.345 8683 

1.448 2982 

1.557 9674 

15 

10 

1.319 9294 

1.372 7857 

1.484 5056 

1.604 7064 

16 

17 

1.343 0281 

1.400 2414 

1.521 6183 

1.652 8476 

17 

IS 

1.366 5311 

1.428 2462 

1.559 6587 

1.702 4331 

18 

10 

1.390 4454 

1.456 8112 

1.598 6502 

1.753 5060 

19 

20 

1.414 7782 

1.485 9474 

1.638 6164 

1.806 1112 

20 

21 

1.439 5368 

1.515 6663 

1.679 5818 

1.860 2946 

21 

2? 

1.464 7287 

1.545 9797 

1.721 5714 

1.916 1034 

22 

23 

1 .490 3615 

1.576 8993 

1 .764 6107 

1 .973 5865 

23 

24 

1.516 4428 

1.608 4372 

1.808 7260 

2.032 7941 

24 

25 

1.542 9805 

1.640 6060 

1 .853 9441 

2.093 7779 

25 

26 

1 .569 9827 

1.673 4181 

1.900 2927 

2.156 5913 

26 

27 

1.597 4574 

1.706 8865 

1 .947 8000 

2.221 2890 

27 

28 

1.625 4129 

1.741 0242 

1.996 4950 

2.287 9277 

28 

29 

1.653 8576 

1.775 8447 

2.046 4074 

2.356 5655 

29 

30 

1.682 8001 

1.811 3616 

2.097 5676 

2.427 2625 

30 

31 

1.712 2491 

1.847 5888 

2.150 0068 

2.500 0804 

31 

32 

1 .742 2135 

1.884 5406 

2.203 7569 

2.575 0828 

32 

33 

1.772 7022 

1.922 2314 

2.258 8509 

2.652 3352 

33 

34 

1 .803 7245 

1.960 6760 

2.315 3221 

2.731 9053 

34 

35 

1.835 2897 

1.999 8896 

2.373 2052 

2.813 8624 

35 

30 

1.867 4073 

2.039 8873 

2.432 5353 

2 .-898 2783 

36 

37 

1.900 0869 

2.080 6851 

2.493 3487 

2.985 2267 

37 

38 

1 .933 3384 

2.122 2988 

2.555 6824 

3.074 7835 

38 

39 

1.967 1718 

2.161 7448 

2.619 5745 

3.167 0270 

39 

40 

2.001 5973 

2.208 0397 

2.685 0638 

3.262 0378 

40 

41 

2.036 6253 

2.252 2005 

2.752 1904 

3.359 8989 

41 

42 

43 

2.072 2662 
2.108 5309 

2.297 2445 
2.343 1894 

2.820 9952 
2.891 5201 

3.460 6959 
3.564 5168 

42 

43 

44 

2.145 4302 

2.390 0531 

2.963 8081 

3.671 4523 

44 

45 

2.182 9752 

2.437 8542 

3.037 9033 

3.781 5958 

45 

40 

2 221 1773 

2.486 6113 

3.113 8509 

3.895 0437 

46 

47 

2.260 0479 

2.536 3435 

3.191 6971 

4.011 8950 

47 

48 

2.299 5987 

2.587 0704 

3.271 4896 

4.132 2519 

48 

49 

2 339 8417 

2.638 8118 

3.353 2768 

4.256 2194 

49 

50 

2.380 7889 

2.691 5880 

3.437 1087 

4.383 9060 

50 

60 

2 831 8163 

3.281 0308 

4.399 7898 

5.891 6031 

60 

70 

3 368 2883 

3.999 5582 

5.632 1029 

7.917 8219 

70 

80 

90 

4.006 3919 

4 765 3808 

4.875 4392 
5.943 1331 

7.209 5678 
9.228 8563 

10.640 8906 
14.300 4671 

SO 

90 

100 

5.668 1559 

7.244 6461 

11.813 7164 

19.218 6320 

100 















94 


TABLES 


TABLE II—AMOUNT OF 1 —Continued 

s = (1 + i) n 


n 

a%% 

4% 

4i/ 2 % 

4 3 / 4 % 

n 

1 

1.035 0000 

1.040 0000 

1.045 0000 

1 .047 5000 

1 

2 

1.071 2250 

1.081 6000 

1.092 0250 

1.097 2562 

2 

3 

1.108 7179 

1 . 124 8640 

1.141 1661 

1.149 3759 

3 

4 

1.147 5230 

1.169 8586 

1 .192 5186 

1.203 9713 

4 

5 

1.187 6863 

1.216 6529 

1.246 1819 

1.261 1599 

5 

6 

1.229 2553 

1.265 3190 

1 .302 2601 

1.321 0650 

6 

7 

1.272 2793 

1.315 9318 

1.360 8618 

1.383 8156 

7 

8 

1.316 8090 

1.368 5690 

1.422 1006 

1.449 5468 

8 

9 

1.362 8974 

1.423 3118 

1.486 0951 

1.518 4003 

9 

10 

1.410 5988 

1.480 2443 

1.552 9694 

1.590 5243 

10 

11 

1.459 9697 

1.539 4541 

1.622 8530 

1 .666 0742 

11 

12 

1.511 0687 

1.601 0322 

1 .695 8814 

1.745 2128 

12 

13 

1.563 9561 

1 .865 0735 

1.772 1961 

1.828 1104 

13 

14 

1.618 6945 

1.731 6764 

1.851 9449 

1.914 9456 

14 

15 

1.675 3488 

1.800 9435 

1.935 2824 

2.005 9055 

15 

16 

1.733 9860 

1.872 9812 

2.022 3702 

2.101 1860 

16 

17 

1 .794 6756 

1.947 9005 

2.113 3768 

2.200 9924 

17 

18 

1.857 4892 

2.025 8165 

2.208 4788 

2.305 5395 

18 

19 

1.922 5013 

2.106 8492 

2.307 8603 

2.415 0526 

19 

20 

1.989 7889 

2.191 1231 

2.411 7140 

2.529 7676 

20 

21 

2.059 4315 

2.278 7681 

2.520 2412 

2.649 9316 

21 

22 

2.131 5116 

2.369 9188 

2.633 6520 

2.775 8034 

22 

23 

2.206 1145 

2.464 7155 

2.752 1664 

2.907 6540 

23 

24 

2.283 3285 

2.563 3042 

2.876 0138 

3.045 7676 

24 

25 

2.363 2450 

2.665 8363 

3.005 4345 

3.190 4415 

25 

26 

2.445 9586 

2.772 4698 

3.140 6790 

3.341 9875 

26 

27 

2.531 5671 

2.883 3686 

3.282 0096 

3.500 7319 

27 

28 

2.620 1720 

2.998 7033 

3.429 7000 

3.667 0167 

28 

29 

2.711 8780 

3.118 6514 

3.584 0365 

3.841 2000 

29 

30 

2.806 7937 

3.243 3975 

3.745 3181 

4.023 6570 

30 

31 

2.905 0315 

3.373 1334 

3.913 8574 

4.214 7807 

31 

32 

3.006 7076 

3.508 0588 

4.089 9810 

4.414 9828 

32 

33 

3.111 9424 

3.648 3811 

4.274 0302 

4.624 6944 

33 

34 

3.220 8603 

3.794 3163 

4.466 3615 

4.844 3674 

34 

35 

3.333 5904 

3.946 0890 

4.667 3478 

5.074 4749 

35 

36 

3.450 2661 

4.103 9326 

4.877 3785 

5.315 5124 

36 

37 

3.571 0254 

4.268 0899 

5.096 8605 

5.567 9993 

37 

38 

3.696 0113 

4.438 8134 

5.326 2192 

5.832 4792 

38 

39 

3.825 3717 

4.616 3660 

5.565 8991 

6.109 5220 

39 

40 

3.959 2597 

4.801 0206 

5.816 3645 

6.399 7243 

40 

41 

4.097 8338 

4.993 0614 

6.078 1009 

6.703 7112 

41 

42 

4.241 2580 

5.192 7839 

6.351 6155 

7.022 1375 

42 

43 

4.389 7020 

5.400 4953 

6.637 4382 

7.355 6890 

43 

44 

4.543 3416 

5.616 5151 

6.936 1229 

7.705 0843 

44 


4.702 3586 

5.841 1757 

7.248 2484 

8.071 0758 

45 

46 

4.866 9411 

6.074 8227 

7.574 4196 

8.454 4519 

46 

47 

5.037 2840 

6.317 8156 

7.915 2685 

8.856 0383 

47 

48 

5.213 5890 

6.570 5282 

8.271 4556 

9.276 7001 

48 

49 

5.396 0646 

6.833 3494 

8.643 6711 

9.717 3434 

49 

50 

5.584 9269 

7.106 6834 

9.032 6363 

10.178 9172 

50 

60 

7.878 0909 

10.519 6274 

14.027 4079 

16.189 8154 

60 

70 

11.112 8253 

15.571 6184 

21.784 1356 

25.750 2954 

70 

80 

15.675 7375 

23.049 7991 

33.830 0964 

40.956 4712 

80 

90 

22.112 1760 

34.119 3333 

52.537 1053 

65.142 2639 

90 

100 

31.191 4080 

50.504 9482 

81 .588 5180 

103.610 3555 

100 




























AMOUNT OF 1 


95 


TABLE II—AMOUNT OF 1 —Continued 

s = (1 + i) n 


n 

5% 

6% 

7 % 

8% 

n 

1 

1.050 0000 

1.060 0000 

1.070 0000 

1.080 0000 

1 

2 

1 .102 5000 

1.123 6000 

1 . 144 9000 

1.166 4000 

2 

3 

1.157 6250 

1 .191 0160 

1.225 0430 

1.259 7120 

3 

4 

1.215 5062 

1.262 4770 

1.310 7960 

1.360 4890 

4 

5 

1.276 2816 

1.338 2256 

1 .402 5517 

1.469 3281 

5 

6 

1.340 0956 

1.418 5191 

1.500 7304 

1.586 8743 

6 

7 

1.407 1004 

1.503 6303 

1.605 7815 

1.713 8243 

7 

8 

1.477 4554 

1 .593 8481 

1.718 1862 

1.850 9302 

8 

9 

1.551 3282 

1.689 4790 

1.838 4592 

1.999 0046 

9 

10 

1.628 8946 

1.790 8477 

1.967 1514 

2.158 9250 

10 

11 

1.710 3394 

1 .898 2986 

2.104 8520 

2.331 6390 

11 

12 

1 .795 8563 

2.012 1965 

2.252 1916 

2.518 1701 

12 

13 

1 .885 6491 

2.132 9283 

2.409 8450 

2.719 6237 

13 

14 

1.979 9316 

2.260 9040 

2.578 5342 

2.937 1936 

14 

15 

2.078 9282 

2.396 5582 

2.759 0315 

3.172 1691 

15 

10 

2.182 8746 

2.540 3517 

2.952 1638 

3.425 9426 

16 

17 

2.292 0183 

2.692 7728 

3.158 8152 

3.700 0180 

17 

18 

2.406 6192 

2.854 3392 

3.379 9323 

3.996 0195 

18 

19 

2.526 9502 

3.025 5995 

3.616 5275 

4.315 7011 

19 

20 

2.653 2977 

3.207 1355 

3.869 6845 

4.660 9571 

20 

21 

2.785 9626 

3.399 5636 

4.140 5624 

5.033 8337 

21 

22 

2.925 2607 

3.603 5374 

4.430 4017 

5.436 5404 

22 

23 

3.071 5238 

3.819 7497 

4.740 5299 

5.871 4636 

23 

24 

3.225 0999 

4.048 9346 

5.072 3670 

6.341 1807 

24 

25 

3.386 3549 

4.291 8707 

5.427 4326 

6.848 4752 

25 

20 

3.555 6727 

4.549 3830 

5.807 3529 

7.396 3532 

26 

27 

3.733 4563 

4.822 3459 

6 213 8676 

7.988 0615 

27 

28 

3.920 1291 

5.111 6867 

6.648 8384 

8.627 1064 

28 

29 

4.116 1356 

5.418 3879 

7.114 2570 

9.317 2749 

29 

30 

4.321 9424 

5.743 4912 

7.612 2550 

10.062 6569 

30 

31 

4.538 0395 

6.088 1006 

8.145 1129 

10.867 6694 

31 

32 

4.764 9415 

6.453 3867 

8.715 2708 

11.737 0830 

32 

33 

5.003 1885 

6.840 5899 

9.325 3398 

12.676 0496 

33 

34 

5.253 3480 

7.251 0253 

9.978 1135 

13.690 1336 

34 

35 

5.516 0154 

7.686 0868 

10.676 5815 

14.785 3443 

35 

30 

5.791 8161 

8.147 2520 

11.423 9422 

15.968 1718 

36 

37 

6.081 4069 

8.636 0871 

12.223 6181 

17.245 6256 

37 

38 

6.385 4773 

9.154 2524 

13.079 2714 

18.625 2756 

38 

39 

6.704 7512 

9.703 5075 

13.994 8204 

20.115 2977 

39 

40 

7.039 9887 

10.285 7179 

14.974 4578 

21.724 5215 

40 

41 

7.391 9882 

10.902 8610 

16.022 6699 

23.462 4832 

41 

42 

7.761 5876 

11.557 0327 

17.144 2568 

25.339 4819 

42 

43 

8.149 6669 

12.250 4546 

18.344 3548 

27.366 6404 

43 

44 

8.557 1503 

12.985 4819 

19.628 4596 

29.555 9717 

44 

45 

8.985 0078 

13.764 6108 

21.002 4518 

31.920 4494 

45 

40 

9.434 2582 

14.590 4875 

22.472 6234 

34.474 0853 

46 

47 

9.905 9711 

15.465 9167 

24.045 7070 

37.232 0122 

47 

48 

10.401 2696 

16.393 S717 

25.728 9065 

40.210 5731 

48 

49 

10.921 3331 

17.377 5040 

27.529 9300 

43.427 4190 

49 

50 

11.467 3998 

18.420 1543 

29.547 0251 

46.901 6125 

50 

00 

18.679 1859 

32.987 6908 

57.946 4268 

101.257 0637 

60 

70 

30.426 4255 

59.075 9302 

113.989 3922 

218.606 4059 

70 

80 

49.561 4411 

105.795 9935 

224.234 3876 

471 .954 8343 

80 

90 

80.730 3650 

189.464 5112 

441.102 9799 

1018.915 0893 

90 

100 

131.501 2578 

339.302 0835 

867.716 3256 

2199.761 2563 

100 















96 


TABLES 


TABLE III—PRESENT VALUE OF 1 
v n = (1 -f i)~ n 


n 

%% 

1 07 

A /o 

1%% 

i y 2 % 

n 

1 

0.995 0249 

0.990 0990 

0.987 6543 

0.985 2217 

1 

o 

fm/ 

0.990 0745 

0.980 2960 

0.975 4611 

0.970 6618 

2 

3 

0.985 1488 

0.970 5902 

0.963 4183 

0.956 3170 

3 

4 

0.980 2475 

0.960 9803 

0.951 5243 

0.942 1842 

4 

5 

0.975 3707 

0.951 4657 

0.939 7771 

0.928 2603 

5 

6 

0.970 5181 

0.942 0452 

0.928 1749 

0.914 5422 

6 

7 

0.965 6896 

0.932 7180 

0.916 7159 

0.901 0268 

7 

8 

0.960 8852 

0.923 4832 

0.905 3984 

0.887 7111 

8 

9 

0.956 1047 

0.914 3398 

0.894 2207 

0.874 5922 

9 

10 

0.951 3479 

0.905 2870 

0.883 1809 

0.861 6672 

10 

11 

0.946 6149 

0.896 3237 

0.872 2775 

0.848 9332 

11 

12 

0.941 9053 

0.887 4492 

0.861 5086 

0.836 3874 

12 

13 

0.937 2192 

0.878 6626 

0.850 8727 

0.824 0270 

13 

11 

0.932 5565 

0.869 9630 

0.840 3681 

0.811 8493 

14 

15 

0.927 9169 

0.861 3495 

0.829 9932 

0.799 8515 

15 

16 

0.923 3004 

0.852 8213 

0.819 7464 

0.788 0310 

16 

17 

0.918 7068 

0.844 3775 

0.809 6260 

0.776 3853 

17 

18 

0.914 1362 

0.836 0173 

0.799 6306 

0.764 9116 

18 

19 

0.909 5882 

0.827 7399 

0.789 7587 

0.753 6075 

19 

20 

0.905 0629 

0.819 5445 

0.780 0086- 

0.742 4704 

20 

21 

0.900 5601 

0.811 4302 

0.770 3788 

0.734. 4980 

21 

22 

0.896 0797 

0.803 3962 

0.760 8680 

0.720 6876 

22 

23 

0.891 6216 

0.795 4418 

0.751 4745 

0.710 0371 

23 

24 

0.887 1857 

0.787 5661 

0.742 1971 

0.699 5439 

24 

25 

0.882 7718 

0.779 7684 

0.733 0341 

0.689 2058 

25 

26 

0.878 3799 

0.772 0480 

0.723 9843 

0.679 0205 

26 

27 

0.874 0099 

0.764 4039 

0.715 0463 

0.668 9857 

27 

28 

0.869 6616 

0.756 8356 

0.706 2185 

0.659 0992 

28 

29 

0.865 3349 

0.749 3422 

0.697 4998 

0.649 3589 

29 

30 

0.861 0297 

0.741 9229 

0.688 8887 

0.639 7624 

30 

31 

0.856 7460 

0.734 5772 

0.680 3839 

0.630 3078 

31 

32 

0.852 4836 

0 727 3041 

0.671 9841 

0.620 9929 

32 

33 

0.848 2424 

0.720 1031 

0.663 6880 

0.611 8157 

33 

34 

0.844 0223 

0.712 9733 

0.655 4943 

0.602 7741 

34 

35 

0.839 8231 

0.705 9142 

0.647 4018 

0.593 8661 

35 

36 

0.835 6449 

0.698 9250 

0.639 4092 

0.585 0897 

36 

37 

0.831 4875 

0.692 0049 

0.631 5152 

0.576 4431 

37 

38 

0.827 3507 

0.685 1534 

0.623 7187 

0.567 9242 

38 

39 

0.823 2346 

0.678 3697 

0.616 0185 

0.559 5313 

39 

10 

0.819 1389 

0.671 6531 

0.608 4133 

0.551 2623 

40 

41 

0.815 0635 

0.665 0031 

0.600 9021 

0.543 1156 

41 

42 

0.811 0085 

0.658 4189 

0.593 4835 

0.535 0892 

42 

43 

0.806 9736 

0.651 8999 

0.586 1566 

0.527 1815 

43 

44 

0.802 9588 

0.645 4455 

0.578 9201 

0.519 3907 

44 

45 

0.798 9640 

0.639 0549 

0.571 7729 

0.511 7149 

45 

46 

0.794 9891 

0.632 7276 

0.564 7140 

0.504 1526 

46 

47 

0.791 0339 

0.626 4630 

0.557 7422 

0.496 7021 

47 

48 

0.787 0984 

0.620 2604 

0.550 8565 

0.489 3617 

48 

49 

0.783 1825 

0.614 1192 

0.544 0558 

0.482 1298 

49 

50 

0.779 2861 

0.608 0388 

0.537 3390 

0.475 0047 

50 

60 

0.741 3722 

0.550 4496 

0.474 5676 

0.409 2960 

60 

70 

0.705 3029 

0.498 3149 

0.419 1290 

0.352 6769 

70 

80 

0.670 9885 

0.451 1179 

0.370 1668 

0.303 8902 

80 

90 

0.638 3435 

0.408 3912 

0.326 9242 

0.261 8522 

90 

100 

0.607 2868 

0.369 7112 

0.288 7333 

0.225 6294 

100 

















PRESENT VALUE OF 1 


97 


TABLE III—PRESENT VALUE OF l—Continued 
v n = (I + i)~ n 


n 

l 3 /4% 

2% 

2 y 2 % 

3% 

n 

1 

0.982 8010 

0.980 3922 

0.975 6098 

0.970 8738 

1 

2 

0.965 8978 

0.961 1688 

0.951 8144 

0.942 5959 

2 

3 

0.949 2853 

0.942 3223 

0.928 5994 

0.915 1417 

3 

4 

0.932 9585 

0.923 8454 

0.905 9506 

0.888 4870 

4 

5 

0.916 9125 

0.905 7308 

0.883 8543 

0.862 6088 

5 

6 

0.901 1425 

0.887 9714 

0.862 2969 

0.837 4843 

6 

7 

0.885 6438 

0.870 5602 

0.841 2652 

0.813 0915 

7 

8 

0.870 4116 

0.853 4904 

0.820 7466 

0.789 4092 

8 

9 

0.855 4414 

0.836 7553 

0.800 7284 

0.766 4167 

9 

10 

0.840 7286 

0.820 3483 

0.781 1984 

0.744 0939 

10 

11 

0.826 2689 

0.804 2630 

0.762 1448 

0.722 4213 

11 

12 

0.812 0579 

0.788 4932 

0.743 5559 

0.701 3799 

12 

13 

0.798 0913 

0.773 0325 

0.725 4204 

0.680 9513 

13 

14 

0.784 3649 

0.757 8750 

0.707 7272 

0.661 1178 

14 

15 

0.770 8746 

0.743 0147 

0.690 4656 

0.641 8620 

15 

16 

0.757 6163 

0.728 4458 

0.673 6249 

0.623 1669 

16 

17 

0.744 5860 

0.714 1626 

0.657 1951 

0.605 0164 

17 

18 

0.731 7799 

0.700 1594 

0.641 1659 

0.587 3946 

18 

19 

0.719 1940 

0.686 4308 

0.625 5277 

0.570 2860 

19 

20 

0.706 8246 

,0.672 9713 

0.610 2709 

0.553 6758 

20 

21 

0.694 6679 

0.659 7758 

0.595 3863 

0.537 5493 

21 

22 

0.682 7203 

0.646 8390 

0.580 8647 

0.521 8925 

22 

23 

0.670 9782 

0.634 1559 

0.566 6972 

0.506 6918 

23 

24 

0.659 4380 

0.621 7215 

0.552 8754 

0.491 9337 

24 

25 

0.648 0963 

0.609 5309 

0.539 3906 

0.477 6056 

25 

26 

0.636 9497 

0.597 5793 

0.526 2347 

0.463 6947 

26 

27 

0.625 9948 

0.585 8620 

0.513 3997 

0.450 1891 

27 

28 

0.615 2283 

0.574 3746 

0.500 8778 

0.437 0768 

28 

29 

0.604 6470 

0.563 1123 

0.488 6612 

0.424 3464 

29 

30 

0.594 2476 

0.552 0709 

0.476 7427 

0.411 9868 

30 

31 

0.584 0272 

0.541 2460 

0.465 1148 

0.399 9872 

31 

32 

0.573 9825 

0.530 6333 

0.453 7706 

0.388 3370 

32 

33 

0.564 1105 

0.520 2287 

0.442 7030 

0.377 0262 

33 

34 

0.554 4084 

0.510 0282 

0.431 9053 

0.366 0449 

34 

35 

0.544 8731 

0.500 0276 

0.421 3711 

0.355 3834 

35 

36 

0.535 5018 

0.490 2232 

0.411 0937 

0.345 0324 

36 

37 

0.526 2917 

0.480 6109 

0.401 0670 

0.334 9829 

37 

38 

0.517 2400 

0.471 1872 

0.391 2849 

0.325 2262 

38 

39 

0.508 3440 

0.461 9482 

0.381 7414 

0.315 7536 

39 

40 

0.499 6010 

0.452 8904 

0.372 4306 

0.306 5568 

40 

41 

0.491 0083 

0.444 0102 

0.363 3470 

0.297 6280 

41 

42 

0.482 5635 

0.435 3041 

0.354 4848 

0.288 9592 

42 

43 

0.474 2639 

0.426 7688 

0.345 8389 

0.280 5429 

43 

44 

0.466 1070 

0.418 4007 

0.337 4038 

0.272 3718 

44 

45 

0.458 0904 

0.410 1968 

0.329 1744 

0.264 4386 

45 

46 

0.450 2117- 

0.402 1537 

0.321 1458 

0.256 7365 

46 

47 

0.442 4685 

0.394 2684 

0.313 3129 

0.249 2588 

47 

48 

0.434 8585 

0.386 5376 

0.305 6712 

0.241 9988 

48 

49 

0.427 3793 

0.378 9584 

0.298 2158 

0.234 9503 

49 

50 

0.420 0288 

0.371 5279 

0.290 9422 

0.228 1071 

50 

60 

0.353 1302 

0.304 7823 

0.227 2836 

0.169 7331 

60 

70 

0.296 8867 

0.250 0276 

0.177 5536 

0.126 2974 

70 

80 

0.249 60 1 1 

0.205 1097 

0.138 7046 

0.093 9771 

80 

90 

0.209 8468 

0.168 2614 

0.108 3558 

0.069 9278 

90 

100 

0.176 4242 

0.138 0330 

0.084 6474 

0.052 0328 

100 



















98 


TABLES 


TABLE III—PRESENT VALUE OF 1 —Continued 
v n - (1 + i)~ n 


n 

3 V 2 V 0 

4% 

4 %% 

4 3 / 4 % 

Tl 

1 

0.966 1836 

0.961 5385 

0.956 9378 

0.954 6539 

1 

2 

0.933 5107 

0.924 5562 

0.915 7300 

0.911 3641 

2 

3 

0.901 9427 

0.888 9964 

0.876 2966 

0.870 0374 

3 

4 

0.871 4422 

0.854 8042 

0.838 5613 

0.830 5846 

4 

5 

0.841 9732 

0.821 9271 

0.802 4510 

0.792 9209 

5 

6 

0.813 5006 

0.790 3145 

0.767 8957 

0.756 9650 

6 

7 

0.785 9910 

0.759 9178 

0.734 8285 

0.722 6396 

7 

8 

0.759 4116 

0.730 6902 

0.703 1851 

0.689 8708 

8 

9 

0.733 7310 

0.702 5867 

0.672 9044 

0.658 5878 

9 

10 

0.708 9188 

0.675 564? 

0.643 9277 

0.628 7235 

10 

11 

0.684 9457 

0.649 5809 

0.616 1987 

0.600 2134 

11 

12 

0.661 7833 

0.624 5970 

0.589 6639 

0.572 9960 

12 

13 

0.639 4042 

0.600 5741 

0.564 2716 

0.547 0129 

13 

14 

0.617 7818 

0.577 4751 

0.539 9729 

0.522 2080 

14 

15 

0.596 8906 

0.555 2645 

0.516 7204 

0.498 5280 

15 

16 

0.576 7059 

0.533 9082 

0.494 4693 

0.475 9217 

16 

17 

0.557 2038 

0.513 3732 

0.473 1764 

0.454 3405 

17 

18 

0.538 3611 

0.493 6281 

0.452 8004 

0.433 7380 

18 

19 

0.520 1557 

0.474 6424 

0.433 3018 

0.414 0696 

19 

20 

0.502 5659 

0.456 3870 

0.414 6429 

0.395 2932 

20 

21 

0.485 5709 

0.438 8336 

0.396 7874 

0.377 3682 

21 

9*) 

0.469 1506 

0.421 9554 

0.379 7009 

0.360 2561 

22 

23 

0.453 2856 

0.405 7263 

0.363 3501 

0.343 9199 

23 

24 

0.437 9571 

0.390 1215 

0.347 7035 

0.328 3245 

24 

25 

0.423 1470 

0.375 1168 

0.332 7306 

0.313 4362 

25 

26 

0.408 8377 

0.360 6892 

0.318 4025 

0.299 2231 

26 

27 

0.395 0122 

0.346 8166 

0.304 6914 

0.285 6546 

27 

28 

0.381 6543 

0.333 4775 

0.291 5707 

0.272 7012 

28 

29 

0.368 7482 

0.320 6514 

0.279 0150 

0.260 3353 

29 

30 

0.356 2784 

0.308 3187 

0.267 0000 

0.248 5301 

30 

31 

0.344 2304 

0.296 4603 

0.255 5024 

0.237 2603 

31 

32 

0.332 5897 

0.285 0579 

0.244 4999 

0.226 5014 

32 

33 

0.321 3427 

0.274 0942 

0.233 9712 

0.216 2305 

33 

34 

0.310 4760 

0.263 5521 

0.223 8959 

0.206 4253 

34 

35 

0.299 9769 

0.253 4155 

0.214 2544 

0.197 0647 

35 

36 

0.289 8327 

0.243 6687 

0.205 0282 

0.188 1286 

36 

37 

0.280 0316 

0.234 2968 

0.196 1992 

0.179 5977 

37 

38 

0.270 5619 

0.225 2854 

0.187 7504 

0.171 4537 

38 

39 

0.261 4125 

0.216 6206 

0.179 6655 

0.163 6789 

39 

40 

0.252 5725 

0.208 2890 

0.171 9287 

0.156 2567 

40 

41 

0.244 0314 

0.200 2779 

0.164 5251 

0.149 1711 

41 

42 

0.235 7791 

0.192 5749 

0.157 4403 

0.142 4068 

42 

43 

0.227 8059 

0.185 1682 

0.150 6605 

0.135 9492 

43 

44 

0.220 1023 

0.178 0464 

0.144 1728 

0.129 7844 

44 

45 

0.212 6592 

0.171 1984 

0.137 9644 

0.123 8992 

45 

46 

0.205 4679 

0.164 6139 

0.132 0233 

0.118 2809 

46 

47 

0.198 5197 

0.158 2826 

0.126 3381 

0.112 9173 

47 

48 

0.191 8064 

0.152 1948 

0.120 8977 

0.107 7970 

48 

49 

0.185 3202 

0.146 3411 

0.115 6916 

0.102 9088 

49 

50 

0.179 0534 

0.140 7126 

0.110 7096 

0.098 2423 

50 

60 

0.126 9343 

0.095 0604 

0.071 2890 

0.061 7672 

60 

70 

0.089 9861 

0.064 2194 

0.045 9050 

0.038 8345 

70 

80 

0.063 7928 

0.043 3843 

0.029 5595 

0.024 4162 

80 

90 

0.045 2240 

0.029 3089 

0.019 0342 

0.015 3510 

90 

100 

0.032 0601 

0.019 8000 

0.012 2566 

0.009 6515 

100 



















PRESENT VALUE OF 1 


99 


TABLE III—PRESENT VALUE OF 1 —Continued 


v n = (1 + i)~ n 


n 

5% 

6% 

7% 

8% 

n 

1 

0.952 3810 

0.943 3962 

0.934 5794 

0.925 9259 

1 

2 

0.907 0295 

0.889 9964 

0.873 4387 

0.857 3388 

2 

3 

0.863 8376 

0.839 6193 

0.816 2979 

0.793 8322 

3 

4 

0.822 7025 

0.792 0937 

0.762 8952 

0.735 0298 

4 

5 

0.783 5262 

0.747 2582 

0.712 9862 

0.680 5832 

5 

6 

0.746 2154 

0.704 9605 

0.666 3422 

0.630 1696 

6 

7 

0.710 6813 

0.665 0571 

0.622 7497 

0.583 4904 

7 

8 

0.676 8394 

0.627 4124 

0.582 0091 

0.540 2689 

8 

9 

0.644 6089 

0.591 8985 

0.543 9337 

0.500 2490 

9 

10 

0.613 9132 

0.558 3948 

0.508 3493 

0.463 1935 

10 

11 

0.584 6793 

0.526 7875 

0.475 0928 

0.428 8829 

11 

12 

0.556 8374 

0.496 9694 

0.444 0120 

0.397 1138 

12 

13 

0.530 3214 

0.468 8390 

0.414 9644 

0.367 6979 

13 

14 

0.505 0680 

0.442 3010 

0.387 8172 

0.340 4610 

14 

15 

0.481 0171 

0.417 2651 

0.362 4460 

0.315 2417 

15 

16 

0.458 1115 

0.393 6463 

0.338 7346 

0.291 8905 

16 

17 

0.436 2967 

0.371 3644 

0.316 5744 

0.270 2690 

17 

18 

0.415 5206 

0.350 3438 

0.295 8639 

0.250 2490 

18 

19 

0.395 7340 

0.330 5130 

0.276 5083 

0.231 7121 

19 

20 

0.376 8895 

0.311 8047 

0.258 4190 

0.214 5482 

20 

21 

0.358 9424 

0.294 1554 

0.241 5131 

0.198 6558 

21 

22 

0.341 8499 

0.277 5051 

0.225 7132 

0.183 9405 

22 

23 

0.325 5713 

0.261 7973 

0.210 9469 

0.170 3153 

23 

24 

0.310 0679 

0.246 9786 

0.197 1466 

0.157 6993 

24 

25 

0.295 3028 

0.232 9986 

0.184 2492 

0.146 0179 

25 

26 

0.281 2407 

0.219 8100 

0.172 1955 

0.135 2018 

26 

27 

0.267 8483 

0.207 3680 

0.160 9304 

0.125 1868 

27 

28 

0.255 0936 

0.195 6301 

0.150 4022 

0.115 9137 

28 

29 

0.242 9463 

0.184 5567 

0.140 5628 

0.107 3275 

29 

30 

0.231 3774 

0.174 1101 

0.131 3671 

0.099 3773 

30 

31 

0.220 3595 

0.164 2548 

0.122 7730 

0.092 0160 

31 

32 

0.209 8662 

0.154 9574 

0.114 7411 

0.085 2000 

32 

33 

0.199 8725 

0.146 1862 

0.107 2347 

0.078 8889 

33 

34 

0.190 3548 

0.137 9115 

0.100 2193 

0.073 0453 

34 

35 

0.181 2903 

0.130 1052 

0.093 6629 

0.067 6345 

35 

36 

0.172 6574 

0.122 7408 

0.087 5355 

0.062 6246 

36 

37 

0.164 4356 

0.115 7932 

0.081 8088 

0.057 9857 

37 

38 

0.156 6054 

0.109 2388 

0.076 4569 

0.053 6905 

38 

39 

0.149 1480 

0.103 0555 

0.071 4550 

0.049 7134 

39 

40 

0.142 0457 

0.097 2222 

0.066 7804 

0.046 0309 

40 

41 

0.135 2816 

0.091 7190 

0.062 4116 

0.042 6212 

41 

42 

0.128 8396 

0.086 5274 

0.058 3286 

0.039 4641 

42 

43 

0.122 7044 

0.081 6296 

0.054 5127 

0.036 5408 

43 

44 

0.116 8613 

0.077 0091 

0.050 9464 

0.033 8341 

44 

45 

0.111 2965 

0.072 6501 

0.047 6135 

0.031 3279 

45 

46 

0.105 9967 

0.068 5378 

CU044 4986 

0.029 0073 

46 

47 

0.100 9492 

0.064 6583 

0.041 5875 

0.026 8586 

47 

48 

0.096 1421 

0.060 9984 

0.038 8668 

0.024 8691 

48 

49 

0.091 5639 

0.057 5457 

0.036 3241 

0.023 0269 

49 

50 

0.087 2037 

0.054 2884 

0.033 9478 

0.021 3212 

50 

60 

0.053 5355 

0.030 3143 

0.017 2573 

0.009 8758 

60 

70 

0.032 8662 

0.016 9274 

0.008 7728 

0.004 5744 

70 

80 

0.020 1770 

0.009 4522 

0.004 4596 

0.002 1188 

80 

90 

0.012 3869 

0.005 2780 

0.002 2670 

0.000 9814 

90 

100 

0.007 6045 

0.002 9472 

0.001 1524 

0.000 4546 

100 


3 

-> ) 


> ) 




















100 


TABLES 


TABLE IV—PRESENT VALUE OF 1 PER ANNUM 


l-v n 

a— =—r— 
n\ l 


tx 

Wo 

1% 

iy 4 % 

1 Wo 

n 

1 

0.995 0249 

0.990 0990 

0.987 6543 

0.985 2217 

l 

2 

1.985 0994 

1.970 3951 

1.963 1154 

1.955 8834 

2 

3 

2.970 2481 

2.940 9852 

2.926 5337 

2.912 2004 

3 

1 

3.950 4957 

3.901 9656 

3.878 0580 

3.854 3846 

4 

5 

4.925 8663 

4.853 4312 

4.817 8350 

4.782 6450 

5 

6 

5.896 3844 

5.795 4765 

5.746 0099 

5.697 1872 

6 

7 

6.862 0740 

6.728 1945 

6.662 7258 

6.598 2140 

7 

8 

7.822 9592 

7.651 6778 

7.568 1243 

7.485 9251 

8 

9 

8.779 0639 

8.566 0176 

8.462 3450 

8.360 5173 

9 

10 

9.730 4119 

9.471 3045 

9.345 5259 

9.222 1846 

10 

11 

10.677 0267 

10.367 6282 

10.217 8034 

10.071 1178 

11 

12 

11.618 9321 

11.255 0775 

11.079 3120 

10.907 5052 

12 

13 

12.556 1513 

12.133 7401 

11.930 1847 

11.731 5322 

13 

14 

13.488 7078 

13.003 7030 

12.770 5528 

12.543 3815 

14 

15 

14.416 6246 

13.865 0525 

13.600 5459 

13.343 2330 

15 

16 

15.339 9250 

14.717 8738 

14.420 2923 

14.131 2640 

16 

17 

16.258 6319 

15.562 2513 

15.229 9183 

14.907 6493 

17 

18 

17.172 7680 

16.398 2686 

16.029 5489 

15.672 5609 

18 

19 

18.082 3562 

17.226 0085 

16.819 3076 

16.426 1684 

19 

20 

18.987 4192 

18.045 5530 

17.599 3161 

17.168 6388 

20 

21 

19.887 9792 

18.856 9831 

18.369 6950 

17.900 1367 

21 

22 

20.784 0590 

19.660 3793 

19.130 5629 

18.620 8244 

22 

23 

21.675 6806 

20.455 8211 

19.882 0374 

19.330 8614 

23 

24 

22.562 8662 

21.243 3873 

20.624 2345 

20.030 4054 

24 

23 

23.445 6380 

22.023 1557 

21.357 2686 

20.719 6112 

25 

26 

24.324 0179 

22.795 2037 

22.081 2530 

21.398 6317 

26 

27 

25.198 0278 

23.559 6076 

22.796 2992 

22.067 6175 

27 

28 

26.067 6894 

24.316 4432 

23.502 5178 

22.726 7167 

28 

29 

26.933 0242 

25.065 7853 

24.200 0176 

23.376 0756 

29 

30 

27.794 0540 

25.807 7082 

24.888 9062 

24.015 8380 

30 

31 

28.650 8000 

26.542 2854 

25.569 2901 

24.646 1458 

31 

32 

29.503 2836 

27.269 5895 

26.241 2742 

25.267 1387 

32 

33 

30.351 5259 

27.989 6926 

26.904 9622 

25.878 9544 

33 

34 

31.195 5482 

28.702 6659 

27.560 4564 

26.481 7285 

34 

35 

32.035 3713 

29.408 5801 

28.207 8582 

27.075 5946 

35 

36 

32.871 0162 

30.107 5050 

28.847 2674 

27.660 6843 

36 

37 

33.702 5037 

30.799 5099 

29.478 7826 

28.237 1274 

37 

38 

34.529 8544 

31.484 6633 

30.102 5013 

28.805 0516 

38 

39 

35.353 0890 

32.163 0330 

30.718 5198 

29.364 5829 

39 

40 

36.172 2279 

32.834 6861 

31.326 9332 

29.915 8452 

40 

41 

36.987 2914 

33.499 6892 

31.927 8352 

30.458 9608 

41 

42 

37.798 2999 

34.158 1081 

32.521 3187 

30.994 0500 

42 

43 

38.605 2735 

34.810 0081 

33.107 4753 

31.521 2316 

43 

44 

39.408 2324 

35.455 4535 

33.686 3954 

32.040 6222 

44 

45 

40.207 1964 

36.094 5084 

34.258 1682 

32.552 3372 

45 

46 

41.002 1855 

36.727 2361 

34.822 8822 

33.056 4898 

46 

47 

41.793 2194 

37.353 6991 

35.380 6244 

33 553 1920 

47 

48 

42.580 3178 

37.973 9595 

35.931 4809 

■ 34 042 5536 

48 

49 

43.363 5003 

38.588 0787 

36.475 5367 

34.524 6834 

49 

£>0 

44.142 7864 

39.196 1175 

37.012 8757 

34.999 6881 

50 

60 

51.725 5608 

44.955 0384 

42.034 5918 

39.380 2689 

60 

<0 

58.939 4176 

50.168 5144 

46.469 6756 

43 154 8718 

70 

80 

65.802 3054 

54.888 2061 

50.386 6571 

46 407 3235 

80 

90 

72.331 2996 

59.160 8815 

53.846 0604 

49 209 8545 

90 

100 | 

78.542 6448 

63.028 8788 

56.901 3394 

51.624 7037 

100 



















PRESENT VALUE OF 1 PER ANNUM 


101 


TABLE IV—PRESENT VALUE OF 1 PER ANNUM —Continued 

I — v n 

a~ = —;— 

n\ i 


n 

l 3 /4% 

2% 

2 y 2 % 

3% 

n 

I 

0.982 8010 

0.980 3922 

0.975 6098 

0.970 8738 

1 

2 

1.948 6988 

1.941 5609 

1 .927 4242 

1.913 4697 

2 

3 

2.897 9840 

2.883 8833 

2.856 0236 

2.828 6114 • 

3 

4 

3.830 9425 

3.807 7287 

3.761 9742 

3.717 0984 

4 

5 

4.747 8551 

4.713 4595 

4.645 8285 

4.579 7072 

5 

e 

5.648 9976 

5.601 4309 

5.508 1254 

5.417 1914 

f> 

7 

6.534 6414 

6.471 9911 

6.349 3906 

6.230 2830 

7 

8 

7.405 0530 

7.325 4814 

7.170 1372 

7.019 6922 

8 

9 

8.260 4943 

8.162 2367 

7.970 8655 

7.786 1089 

9 

10 

9.101 2229 

8.982 5850 

8.752 0639 

8.530 2028 

10 

11 

9.927 4918 

9.786 8480 

9.514 2087 

9.252 6241 

11 

12 

10.739 5497 

10.575 3412 

10.257 7646 

9.954 0040 

12 

13 

11.537 6410 

11.348 3738 

10.983 1850 

10.634 9553 

13 

11 

12.322 0059 

12.106 2488 

11.690 9122 

11 .296 0731 

14 

15 

13.092 8805 

12.849 2635 

12.381 3777 

11 .937 9351 

15 

16 

13.850 4968 

13.577 7093 

13.055 0027 

12.561 1020 

16 

17 

14.595 0828 

14.291 8719 

13.712 1977 

13.166 1185 

17 

18 

15.326 8627 

14.992 0312 

14.353 3636 

13.753 5131 

18 

19 

16.046 0567 

15.678 4620 

14.978 8913 

14.323 7991 

19 

20 

16.752 8813 

16.351 4333 

15.589 1623 

14.877 4749 

20 

21 

17.447 5492 

17.011 2092 

16.184 5486 

15.415 0241 

21 

22 

18.130 2695 

17.658 0482 

16.765 4132 

15.936 9166 

22 

23 

18.801 2476 

18.292 2041 

17.332 1105 

16.443 6084 

23 

24 

19.460 6856 

18.913 9256 

17.884 9858 

16.935 5421 

24 

25 

20.108 7820 

19.523 4565 

18.424 3764 

17.413 1477 

25 

26 

20-.745 7317 

20.121 0358 

18.950 6111 

17.876 8424 

26 

27 

21.371 7264 

20.706 8978 

19.464 0109 

18.327 0315 

27 

28 

21.986 9547 

21.281 2724 

19.964 8887 

18.764 1082 

28 

29 

22.591 6017 

21.844 3847 

20.453 5499 

19.188 4546 

29 

30 

23.185 8493 

22.396 4556 

20.930 2926 

19.600 4414 

30 

31 

23.769 8765 

22.937 7015 

21.395 4074 

20.000 4285 

31 

32 

24.343 8590 

23.468 3348 

21 .849 1780 

20.388 7655 

32 

33 

24.907 9695 

23.988 5636 

22.291 8809 

20.765 7918 

33 

34 

25.462 3779 

24.498 5917 

22.723 7863 

21.131 8367 

34 

35 

26.007 2510 

24.998 6193 

23.145 1573 

21.487 2201 

35 

36 

26.542 7528 

25.488 8425 

23.556 2511 

21.832 2525 

36 

37 

27.069 0446 

25.969 4534 

23.957 3181 

22.167 2354 

37 

38 

27.586 2846 

26.440 6406 

24.348 6030 

22.492 4616 

38 

39 

28.094 6286 

26.902 5888 

24.730 3444 

22.808 2151 

39 

40 

28.594 2296 

27.355 4792 

25.102 7750 

23.114 7720 

40 

41 

29.085 2379 

27.799 1894 

25.466 1220 

23.412 4000 

41 

42 

29.567 8014 

28.234 7936 

25.820 6068 

23.701 3592 

42 

43 

30.042 0652 

28.661 5623 

26.166 4457 

23.981 9021 

43 

44 

30.508 1722 

29.079 9631 

26.503 8494 

24.254 2739 

44 

45 

30.966 2626 

29.490 1599 

26.833 0239 

24.518 7125 

45 

46 

31.416 4743 

29.892 3136 

27.154 1696 

24.775 4491 

46 

47 

31.858 9428 

30.286 5820 

27.467 4826 

25.024 7078 

47 

48 

32.293 8013 

30.673 1196 

27.773 1537 

25.266 7066 

48 

49 

32.721 1806 

31.052 0780 

28.071 3695 

25.501 6569 

49 

50 

33.141 2095 

31.423 6059 

28.362 3117 

25.729 7640 

50 

60 

36 963 9855 

34.760 8867 

30.908 6565 

27.675 5637 

60 

70 

40 177 9027 

37 498 6193 

32.897 8570 

29.123 4214 

70 

80 

42 879 9347 

39.744 5136 

34.451 8172 

30.200 7634 

80 

90 

45 151 6104 

41.586 9292 

35.665 7685 

31 .002 4071 

90 

100 

47.061 4730 

43.098 3516 

36.614 1053 

31.598 9053 

100 































102 


TABLES 


TABLE IV—PRESENT VALUE OF 1 PER ANNUM— Continued^ 

l — v n 


n 

3%% 

4% 

4»/ 2 % 

4 3 /4% 

n 

1 

0.966 1836 

0.961 5385 

0.956 9378 

0.954 6539 

1 

2 

1.899 6943 

1.886 0947 

1 .872 6678 

1.866 0181 

2 

3 

2.801 6370 

2.775 0910 

2.748 9644 

2.736 0554 

3 

4 

3.673 0792 

3.629 8952 

3.587 5257 

3.566 6400 

4 

5 

4.515 0524 

4.451 8223 

4.389 9767 

4.359 5609 

5 

6 

5.328 5530 

5.242 1369 

5.157 8725 

5.116 5259 

6 

7 

6.114 5440 

6.002 0547 

5.892 7009 

5.839 1656 

7 

8 

6.873 9555 

6.732 7449 

6.595 8861 

6.529 0363 

8 

9 

7.607 6865 

7.435 3316 

7.268 7905 

7.187 6242 

9 

10 

8.316 6053 

8.110 8958 

7.912 7182 

7.816 3477 

10 

11 

9.001 5510 

8.760 4767 

8.528 9169 

8.416 5610 

11 

12 

9.663 3343 

9.385 0738 

9.118 5808 

8.989 5571 

12 

13 

10.302 7385 

9.985 6478 

9.682 8524 

9.536 5700 

13 

14 

10.920 5203 

10.563 1229 

10.222 8253 

10.058 7780 

14 

15 

11.517 4109 

11.118 3874 

10.739 5457 

10.557 3060 

15 

16 

12.094 1168 

11.652 2956 

11.234 0150 

11.033 2277 ' 

16 

17 

12.651 3206 

12.165 6688 

11.707 1914 

11.487 5682 

17 

18 

13.189 6817 

12.659 2970 

12.159 9918 

11.921 3062 

18 

19 

13.709 8374 

13.133 9394 

12.593 2936 

12.335 3758 

19 

20 

14.212 4033 

13.590 3263 

13.007 9364 

12.730 6690 

20 

21 

14.697 9742 

14.029 1600 

13.404 7239 

13.108 0372 

21 

22 

15.167 1248 

14.451 1153 

13.784 4248 

13.468 2933 

22 

23 

15.620 4105 

14.856 8417 

14.147 7749 

13.812 2132 

23 

24 

16.058 3676 

15.246 9631 

14.495 4784 

14.140 5376 

24 

25 

16.481 5146 

15.622 0799 

14.828 2090 

14.453 9739 

25 

26 

16.890 3523 

15.982 7692 

15.146 6114 

14.753 1970 

26 

27 

17.285 3645 

16.329 5858 

15.451 3028 

15.038 8516 

27 

28 

17.667 0188 

16.663 0632 

15.742 8735 

15.311 5528 

28 

29 

18.035 7670 

16.983 7146 

16.021 8885 

15.571 8881 

29 

30 

18.392 0454 

17.292 0333 

16.288 8885 

15.820 4183 

30 

31 

18.736 2758 

17.588 4936 

16.544 3910 

16.057 6785 

31 

32 

19.068 8655 

17.873 5515 

16.788 8909 

16.284 1800 

32 

33 

19.390 2082 

18.147 6457 

17.022 8621 

16.500 4105 

33 

34 

19.700 6842 

18.411 1978 

17.246 7580 

16.706 8358 

34 

35 

20.000 6611 

18.664 6132 

17.461 0124 

16.903 9005 

35 

36 

20.290 4938 

18.908 2820 

17.666 0406 

17.092 0291 

36 

37 

20.570 5254 

19.142 5788 

17.862 2398 

17.271 6269 

37 

38 

20.841 0874 

19.367 8642 

18.049 9902 

17.443 0805 

38 

39 

21.102 4999 

19.584 4848 

18.229 6557 

17.606 7595 

39 

40 

21 .355 0723 

19.792 7739 

18.401 5844 

17.763 0162 

40 

41 

21.599 1037 

19.993 0518 

18.566 1095 

17.912 1873 

41 

42 

21.834 8828 

20.185 6267 

18.723 5498 

18.054 5941 

42 

43 

22.062 6887 

20.370 7949 

18.874 2103 

18.190 5433 

43 

44 

22.282 7910 

20.548 8413 

19.018 3830 

18.320 3277 

44 

45 

22.495 4503 

20.720 0397 

19.156 3474 

18.444 2269 

45 

46 

22.700 9181 

20.884 6536 

19.288 3707 

18.562 5078 

46 

47 

22.899 4378 

21.042 9361 

19.414 7088 

18.675 4251 

47 

48 

23.091 2442 

21.195 1309 

19.535 6065 

18.783 2221 

48 

49 

23.276 5645 

21 .341 4720 

19.651 2981 

18.886 1308 

49 

50 

23.455 6179 

21.482 1846 

19.762 0078 

18.984 3731 

50 

60 

24.944 7341 

22.623 4900 

20.638 0220 

19.752 2689 

60 

70 

26.000 3966 

23.394 5150 

21.202 1119 

20.235 0630 

70 

80 

26.748 7757 

23.915 3918 

21.565 3449 

20.538 6070 

80 

90 

27.279 3156 

24.267 2776 

21.799 2408 

20.729 4523 

90 

100 

27.655 4254 

24.504 9990 

21.949 8527 

20.849 4412 

100 















PRESENT VALUE OF 1 PER ANNUM 


103 


TABLE IV—PRESENT VALUE OF 1 PER ANNUM —Continued 

K 

l-V n 

a -= -— 

n\ i 


n 

5% 

6% 

7% 

8% 

n 

I 

0.952 3810 

0.943 3962 

0.934 5794 

0.925 9259 

1 

2 

1.859 4104 

1.833 3927 

1 .808 0182 

1.783 2648 

2 

3 

2.723 2480 

2.673 0120 

2.624 3160 

2.577 0970 

3 

4 

3.545 9505 

3.465 1056 

3.387 2113 

3.312 1268 

4 

5 

4.329 4767 

4.212 3638 

4.100 1974 

3.992 7100 

5 

6 

5.075 6921 

4.917 3243 

4.766 5397 

4.622 8797 

6 

7 

5.786 3734 

5.582 3814 

5.389 2894 

5.206 3701 

7 

8 

6.463 2128 

6.209 7938 

5.971 2985 

5.746 6389 

8 

9 

7.107 8217 

6.801 6923 

6.515 2322 

6.246 8879 

9 

10 

7.721 7349 

7.360 0870 

7,023 5816 

6.710 0814 

10 

11 

8.306 4142 

7.886 8746, 

7.498 6744 

7.138 9643 

11 

12 

8.863 2516 

8.383 8439 

7.942 6863 

7.536 0780 

12 

13 

9.393 5730 

8.852 6830 

8.357 6508 

7.903 7759 

13 

14 

9.898 6409 

9.294 9839 

8.745 4680 

8.244 2370 

14 

15 

10.379 6580 

9.712 2490 

9.107 9140 

8.559 4787 

15 

16 

10.837 7696 

10.105 8953 

9.446 6486 

8.851 3692 

16 

17 

11.274 0662 

10.477 2597 

9.763 2230 

9.121 6381 

17 

18 

11.689 5869 

10.827 6035 

10.059 0869 

9.371 8871 

18 

19 

12.085 3209 

11.158 1165 

10.335 5952 

9.603 5992 

19 

20 

12.462 2103 

11 .469 9212 

10.594 0143 

9.818 1474 

20 

21 

12.821 1527 

11.764 0766 

10.835 5273 

10.016 8032 

21 

22 

13.163 0026 

12.041 5817 

11.061 2405 

10.200 7437 

22 

23 

13.488 5739 

12.303 3790 

11.272 1874 

10.371 0590 

23 

24 

13.798 6418 

12.550 3575 

11.469 3340 

10.528 7583 

24 

25 

14.093 9446 

12.783 3562 

11.653 5832 

10.674 7762 

25 

26 

14.375 1853 

13.003 1662 

11.825 7787 

10.809 9780 

26 

27 

14.643 0336 

13.210 5341 

11.986 7090 

10.935 1648 

27 

28 

14.898 1273 

13.406 1643 

12.137 1113 

11.051 0785 

28 

29 

15.141 0736 

13.590 7210 

12.277 6741 

11.158 4060 

29 

30 

15.372 4510 

13.764 8312 

12.409 0412 

11.257 7833 

30 

31 

15.592 8105 

13.929 0860 

12.531 8142 

11.349 7994 

31 

32 

15.802 6767 

14.084 0434 

12.646 5553 

11.434 9994 

32 

33 

16.002 5492 

14.230 2296 

12.753 7900 

11.513 8884 

33 

34 

16.192 9040 

14.368 1411 

12.854 0094 

11.586 9337 

34 

35 

16.374 1943 

14.498 2464 

12.947 6723 

11.654 5682 

35 

36 

16.546 8517 

14.620 9871 

13.035 2078 

11.717 1928 

36 

37 

16.711 2873 

14.736 7803 

13.117 0166 

11.775 1785 

37 

38 

16.867 8927 

14.846 0192 

13.193 4735 

11.828 8690 

38 

39 

17.017 0407 

14.949 0747 

13.264 9285 

11.878 5824 

39 

40 

17.159 0864 

15.046 2969 

13.331 7088 

11.924 6133 

40 

41 

17.294 3680 

15.138 0159 

13.394 1204 

11.967 2346 

41 

42 

17.423 2076 

15.224 5433 

13.452 4490 

12.006 6t:87 

42 

43 

17.545 9120 

15.306 1729 

13.506 9617 

12.043 2395 

43 

44 

17.662 7733 

15.383 1820 

13.557 9081 

12.077 0736 

44 

45 

17.774 0698 

15.455 8321 

13.605 5216 

12.108 4015 

45 

46 

17.880 0665 

15.524 3699 

13.650 0202 

12.137 4088 

46 

47 

17.981 0157 

15.589 0282 

13.691 6076 

12.164 2674 

47 

48 

18.077 1578 

15.650 0266 

13.730 4744 

12.189 1365 

48 

49 

18.168 7217 

15.707 5723 

13.766 7986 

12.212 1634 

49 

50 

18.255 9255 

15.761 8606 

13.800 7463 

12.233 4846 

50 

60 

18.929 2895 

16.161 4277 

14.039 1812 

12.376 5518 

60 

70 

19.342 6766 

16.384 5439 

14.160 3893 

• 12.442 8196 

70 

80 

1 9.596 4605 

16.509 1308 

14.222 0054 

12.473 5144 

80 

90 

19 752 2617 

16.578 6994 

14.253 3279 

12.487 7320 

90 

100 

19.847 9102 

16.617 5462 

14.269 2507 

12 494 3176 

100 

















104 


TABLES 


TABLE V—AMOUNT OF 1 PER ANNUM 


_ (l+;) n -l 
n\ i 


n 

y 2 % 

1% 

1 Va% 

iy 2 % 

n 

1 

1 .000 0000 

1 .000 0000 

1.000 0000 

1 .000 0000 

1 

o 

h* 

2.005 0000 

2.010 0000 

2.012 5000 

2.015 0000 

2 

3 

3.015 0250 

3.030 1000 

3.037 6562 

3.045 2250 

3 

4 

4.030 1001 

4.060 4010 

4.075 6270 

4.090 9034 

4 

5 

5.050 2506 

5.101 0050 

5.126 5723 

5.152 2669 

5 

6 

6.075 5019 

6.152 0151 

6.190 6544 

6.229 5509 

6 

7 

7.105 8794 

7.213 5352 

7.268 0376 

7.322 9942 

7 

8 

8.141 4088 

8.285 6706 

8.358 8881 

8.432 8391 

8 

9 

9.182 1158 

9.368 5273 

9.463 3742 

9.559 3317 

9 

10 

10.228 0264 

10.462 2125 

10.581 6664 

10.702 7217 

10 

11 

11.279 1665 

11.566 8347 

11.713 9372 

11.863 2625 

11 

12 

12.335 5624 

12.682 5030 

12.860 3614 

13.041 2114 

12 

13 

13.397 2402 

13.809 3280 

14.021 1159 

14.236 8296 

13 

14 

14.464 2264 

14.947 4213 

15.196 3799 

15.450 3820 

14 

15 

15.536 5475 

16.096 8955 

16.386 3346 

16.682 1378 

15 

16 

16.614 2303 

17.257 8645 

17.591 1638 

17.932 3698 

16 

17 

17.697 3014 

18.430 4431 

18.811 0534 

19.201 3554 

17 

18 

18.785 7879 

19.614 7476 

20.046 1915 

20.489 3757 

18 

19 

19.879 7168 

20.810 8950 

21.296 7689 

21.796 7164 

19 

20 

20.979 1154 

22.019 0040 

22.562 9785 

23.123 6671 

20 

21 

22.084 0110 

23.239 1940 

23.845 0158 

24.470 5221 

21 

22 

23.194 4311 

24.471 5860 

25.143 0785 

25.837 5799 

22 

23 

24.310 4032 

25.716 3018 

26.457 3670 

27.225 1436 

23 

24 

25.431 9552 

26.973 4648 

27.788 0840 

28.633 5208 

24 

25 

26.559 1150 

28.243 1995 

29.135 4351 

30.063 0236 

25 

26 

27.691 9106 

29.525 6315 

30.499 6280 

31.513 9690 

26 

27 

28.830 3702 

30.820 8878 

31.880 8734 

32.986 6785 

27 

28 

29.974 5220 

32.129 0967 

33.279 3843 

34.481 4787 

28 

29 

31.124 3946 

33.450 3877 

34.695 3766 

35.998 7008 

29 

30 

32.280 0166 

34 784 8915 

36.129 0688 

37.538 6814 

30 

31 

33.441 4167 

36.132 7404 

37.580 6822 

39.101 7616 

31 

32 

34.608 6238 

37.494 0678 

39.050 4407 

40.688 2880 

32 

33 

35.781 6669 

38.869 0085 

40.538 5712 

42.298 6123 

33 

34 

36.960 5752 

40.257 6986 

42.045 3033 

43.933 0915 

34 

35 

38.145 3781 

41.660 2756 

43.570 8696 

45.592 0879 

35 

36 

39.336 1050 

43.076 8784 

45.115 5055 

47.275 9692 

36 

37 

40.532 7855 

44.507 6471 

46.679 4493 

48.985 1087 

37 

38 

41.735 4494 

45.952 7236 

48.262 9424 

50.719 8854 

38 

39 

42.944 1267 

47.412 2508 

49.866 2292 

52.480 6837 

39 

40 

44.158 8473 

48.886 3734 

51.489 5571 

54.267 8939 

40 

41 

45.379 6415 

50.375 2371 

53.133 1765 

56.081 9123 

41 

42 

46.606 5397 

51.878 9895 

54.797 3412 

57.923 1410 

42 

43 

47.839 5724 

53.397 7794 

56.482 3080 

59.791 9881 

43 

44 

49.078 7703 

54.931 7572 

58.188 3369 

61 .688 8679 

44 

45 

50.324 1642 

56.481 0747 

59.915 6911 

63.614 2010 

45 

46 

51.575 7850 

58.045 8855 

61.664 6372 

65.568 4140 

46 

47 

52.833 6639 

59.626 3443 

63.435 4452 

67.551 9402 

47 

48 

54.097 8322 

61.222 6078 

65.228 3882 

69.565 2193 

48 

49 

55.368 3214 

62.834 8338 

67.043 7431 

71.608 6976 

49 

50 

56.645 1630 

64.463 1822 

68.881 7899 

73.682 8280 

50 

60 

69.770 0305 

81.669 6699 

88.574 5078 

96.214 6517 

60 

70 

83.566 1055 

100.676 3368 

110.871 9978 

122.363 7530 

70 

80 

98.067 7136 

121.671 5217 

136.118 7953 

152.710 8525 

80 

90 

113.310 9358 

144.863 2675 

164.705 0076 

187.929 9004 

90 

100 

129.333 6984 

170.481 3829 

197.072 3420 

228.803 0433 

100 



















AMOUNT OF 1 PER ANNUM 


105 


TABLE Y—AMOUNT OF 1 PER ANNUM —Continued 


s __ (!+;)”-! 
n\ i 


n 

l 3 /4% 

2% 

2V 2 % 

3% 

n 

1 

1.000 0000 

1 .000 0000 

1 .000 0000 

1 .000 0000 

1 

•> 

2.017 5000 

2.020 0000 

2.025 0000 

2.030 0000 

2 

3 

3.052 8062 

3.060 4000 

3.075 6250 

3.090 9000 

3 

4 

4.106 2304 

4.121 6080 

4.152 5156 

4.183 6270 

1 

5 

5.178 0894 

5.204 0402 

5.256 3285 

5.309 1358 

5 

6 

6.268 7060 

6.308 1210 

6.387 7367 

6.468 4099 

6 

7 

7.378 4083 

7.434 2834 

7.547 4302 

7.662 4622 

7 

8 

8.507 5304 

8.582 9690 

8.736 1159 

8.892 3360 

8 

9 

9.656 4122 

9.754 6284 

9.954 5188 

10.159 1061 

9 

10 

10.825 3994 

10.949 7210 

11.203 3818 

11.463 8793 

10 

11 

12.014 8439 

12.168 7154 

12.483 4663 

12.807 7957 

11 

12 

13.225 1037 

13.412 0897 

13.795 5530 

14.192 0296 

12 

13 

14.456 5430 

14.680 3315 

15.140 4408 

15.617 7904 

13 

14 

15.709 5325 

15.973 9382 

16.518 9528 

17.086 3242 

14 

15 

16.984 4494 

17.293 4169 

17.931 9267 

18.598 9139 

15 

10 

18.281 6772 

18.639 2852 

19.380 2248 

20.156 8813 

16 

17 

19.601 6066 

20.012 0710 

20.864 7304 

21.761 5877 

17 

18 

20.944 6347 

21.412 3124 

22.386 3487 

23.414 4354 

18 

19 

22.311 1658 

22.840 5586 

23.946 0074 

25.116 8684 

19 

20 

23.701 6112 

24.297 3698 

25.544 6576 

26.870 3745 

20 

21 

25.116 3894 

25.783 3172 

27.183 2740 

28.676 4857 

21 

•>*) 

26.555 9262 

27.298 9835 

28.862 8559 

30.536 7803 

22 

23 

28.020 6549 

28.844 9632 

30.584 4273 

32.452 8837 

23 

24 

29.511 0164 

30.421 8625 

32.349 0380 

34.426 4702 

24 

25 

31.027 4592 

32.030 2997 

34.157 7639 

36.459 2643 

25 

26 

32.570 4397 

33.670 9057 

36.011 7080 

38.553 0422 

26 

27 

34.140 4224 

35.344 3238 

37.912 0007 

40.709 6335 

27 

28 

35.737 8798 

37.051 2103 

39.859 8008 

42.930 9225 

28 

29 

37.363 2927 

38.792 2345 

41.856 2958 

45.218 8502 

29 

30 

39.017 1503 

40.568 0792 

43.902 7032 

47.575 4157 

30 

31 

40.699 9504 

42.379 4408 

46.000 2707 

50.002 6782 

31 

32 

42.412 1996 

44.227 0296 

48.150 2775 

52.502 7585 

32 

33 

44.154 4130 

46.111 5702 

50.354 0344 

55.077 8413 

33 

34 

45.927 1153 

48.033 8016 

52.612 8853 

57.730 1765 

34 

35 

47.730 8398 

49.994 4776 

54.928 2074 

60.462 0818 

35 

36 

49.566 1295 

51.994 3672 

57.301 4126 

63.275 9443 

36 

37 

51.433 5368 

54.034 2545 

59.733 9479 

66.174 2226 

37 

38 

53.333 6236 

56.114 9396 

62.227 2966 

69.159 4493 

38 

39 

55.266 9621 

58.237 2384 

64.782 9791 

72.234 2328 

39 

40 

57.234 1339 

60.401 9832 

67.402 5535 

75.401 2597 

40 

41 

59.235 7312 

62.610 0228 

70.087 6174 

78.663 2975 

41 

42 

61.272 3565 

64.862 2233 

72.839 8078 

82.023 1964 

42 

43 

63.344 6228 

67.159 4678 

75.660 8030 

85.483 8923 

43 

44 

65.453 1537 

69.502 6571 

78.552 3231 

89.048 4091 

44 

45 

67.598 5839 

71.892 7103 

81.516 1312 

92.719 8614 

45 

46 

69.781 5591 

74.330 5645 

84.554 0344 

96.501 4572 

46 

47 

72.002 7364 

76.817 1758 

87.667 8853 

100.396 5010 

47 

48 

74.262 7842 

79.353 5193 

90.859 5824 

104.408 3960 

48 

49 

76.562 3830 

81.940 5897 

94.131 0720 

108.540 6478 

49 

50 

78.902 2247 

84.579 4014 

97.484 3488 

112.796 8673 — 

50 

60 

104.675 2159 

114.051 5394 

135.991 5900 

163.053 4368 

60 

70 

135.330 7583 

149.977 9111 

185.284 1142 

230.594 0637 

70 

80 

171 .793 8242 

193.771 9578 

248.382 7126 

321.363 0186 

80 

90 

215.164 6172 

247.156 6563 

329.154 2533 

443.348 9036 

90 

100 

266.751 7679 

312.232 3059 

432.548 6540 

607.287 7327 

100 

















106 


TABLES 


TABLE Y—AMOUNT OF 1 PER ANNUM 


(l + f) n -l 
nT i 


n 

3%% 

4% 

4y 2 % 

4 3/4% 

n 

1 

1.000 0000 

1.000 0000 

1 .000 0000 

1 .000 0000 

1 

2 

2.035 0000 

2.040 0000 

2.045 0000 

2.047 5000 

2 

3 

3.106 2250 

3.121 6000 

3.137 0250 

3.144 7562 

3 

4 

4.214 9429 

4.246 4640 

4.278 1911 

4.294 1322 

4 

5 

5.362 4659 

5.416 3226 

5.470 7097 

5.498 1034 

5 

6 

6.550 1522 

6.632 9755 

6.716 8917 

6.759 2634 

6 

7 

7.779 4075 

7.898 2945 

8.019 1518 

8.080 3284 

7 

8 

9.051 6868 

9.214 2263 

9.380 0136 

9.464 1440 

8 

9 

10.368 4958 

10.582 7953 

10.802 1142 

10.913 6908 

9 

10 

11.731 3932 

12.006 1071 

12.288 2094 

12.432 0911 

10 

11 

13.141 9919 

13.486 3514 

13.841 1788 

14.022 6154 

11 

12 

14.601 9616 

15.025 8055 

15.464 0318 

15.688 6897 

12 

13 

16.113 0303 

16.626 8377 

17.159 9133 

17.433 9024 

13 

14 

17.676 9864 

18.291 9112 

18.932 1094 

19.262 0128 

14 

15 

19.295 6809 

20.023 5876 

20.784 0543 

21.176 9584 

15 

16 

20.971 0297 

21.824 5311 

22.719 3367 

23.182 8640 

16 

17 

22.705 0158 

23.697 5124 

24.741 7069 

25.284 0500 

17 

18 

24.499 6913 

25.645 4129 

26.855 0837 

27.485 0424 

18 

19 

26.357 1805 

27.671 2294 

29.063 5625 

29.790 5819 

19 

20 

28.279 6818 

29.778 0786 

31.371 4228 

32.205 6345 

20 

21 

30.269 4707 

31.969 2017 

33.783 1368 

34.735 4022 

21 

22 

32.328 9022 

34.247 9698 

36.303 3780 

37.385 3338 

22 

23 

34.460 4137 

36.617 8886 

38.937 0300 

40.161 1371 

23 

24 

36.666 5282 

39.082 6041 

41.689 1963 

43.068 7S11 

24 

25 

38.949 8567 

41.645 9083 

44.565 2102 

46.114 5587 

25 

26 

41.313 1017 

44.311 7446 

47.570 6446 

49.305 0002 

26 

27 

43.759 0602 

47.084 2144 

50.711 3236 

52.646 9877 

27 

28 

46.290 6273 

49.967 5830 

53.993 3332 

56.147 7197 

28 

29 

48.910 7993 

52.966 2863 

57.423 0332 

59.814 7363 

29 

30 

51.622 6773 

56.084 9378 

61.007 0697 

63.655 9363 

30 

31 

54.429 4710 

59.328 3353 

64.752 3878 

67.679 5933 

31 

32 

57.334 5025 

62.701 4687 

68.666 2452 

71.894 3740 

32 

33 

60.341 2100 

66.209 5274 

72.756 2263 

76.309 3567 

33 

34 

63.453 1524 

69.857 9085 

77.030 2565 

80.934 0512 

34 

35 

66.674 0127 

73.652 2249 

81.496 6180 

85.778 4186 

35 

36 

70.007 6032 

77.598 3138 

86.163 9658 

90.852 8935 

36 

37 

73.457 8693 

81.702 2464 

91.041 3443 

96.168 4059 

37 

38 

77.028 8947 

85.970 3363 

96.138 2048 

101.736 4052 

38 

39 

80.724 9060 

90.409 1497 

101 .464 4240 

107.568 8845 

59 

40 

84.550 2778 

95.025 5157 

107.030 3231 

113.678 4065 

40 

41 

88.509 5375 

99.826 5363 

112.846 6876 

120.078 1308 

41 

42 

92.607 3713 

104.819 5978 

118.924 7885 

126.781 8420 

42 

43 

96.848 6293 

110.012 3817 

125.276 4040 

133.803 9795 

43 

44 

101.238 3313 

115.412 8770 

131.913 8422 

141.159 6685 

44 

45 

105.781 6729 

121.029 3920 

138.849 9651 

148.864 7528 

45 

46 

110.484 0314 

126.870 5677 

146.098 2135 

156.935 8285 

46 

47 

115.350 9726 

132.945 3904 

153.672 6331 

165.390 2804 

47 

48 

120.388 2566 

139.263 2060 

161.587 9016 

174.246 3187 

48 

49 

125.601 8456 

145.833 7343 

169.859 3572 

183.523 0188 

49 

50 

130.997 9102 

152.667 0837 

178.503 0283 

193.240 3622 

50 

60 

196.516 8829 

237.990 6852 

289.497 9540 

319.785 5885 

60 

70 

288.937 8646 

364.290 4588 

461.869 6796 

521 .058 8495 

70 

80 

419.306 7868 

551.244 9768 

729.557 6985 

841.188 8678 

80 

90 

603.205 0270 

827.983 3335 

1145.269 0066 

1350.363 4500 

90 

100 

862.611 6567 

1237.623 7046 

1790.855 9563 

2160.218 0106 

100 

















AMOUNT OF 1 PER ANNUM 


107 


TABLE V—AMOUNT OF 1 PER ANNUM —Continued 


(l + j) n -l 
n| i 


n 

5% 

6% 

7% 

8% 

n 

1 

1.000 0000 

1.000 0000 

1 .000 0000 

1.000 0000 

1 

2 

2.050 0000 

2.060 0000 

2.070 0000 

2.080 0000 

2 

3 

3.152 5000 

3.183 6000 

3.214 9000 

3.246 4000 

3 

4 

4.310 1250 

4.374 6160 

4.439 9430 

4.506 1120 

4 

5 

5.525 6312 

5.637 0930 

5.750 7390 

5.866 6010 

5 

6 

6.801 9128 

6.975 3185 

7.153 2907 

7.335 9290 

6 

7 

8.142 0084 

8.393 8376 

8.654 0211 

8.922 8034 

7 

8 

9.549 1089 

9.897 4679 

10.259 8026 

10.636 6276 

8 

9 

11.026 5643 

11.491 3160 

11.977 9888 

12.487 5578 

9 

10 

12.577 8925 

13.180 7949 

13.816 4480 

14.486 5625 

10 

11 

14.206 7872 

14.971 6426 

15.783 5993 

16.645 4875 

11 

12 

15.917 1265 

16.869 9412 

17.888 4513 

18.977 1265 

12 

13 

17.712 9828 

18.882 1377 

20.140 6429 

21.495 2966 

13 

14 

19.598 6320 

21.015 0659 

22.550 4879 

24.214 9203 

14 

15 

21.578 5636 

23.275 9699 

25.129 0220 

27.152 1139 

15 

16 

23.657 4918 

25.672 5281 

27.888 0536 

30.324 2830 

16 

17 

25.840 3664 

28.212 8798 

30.840 2173 

33.750 2257 

17 

18 

28.132 3847 

30.905 6526 

33.999 0325 

37.450 2437 

18 

19 

30.539 0039 

- 33.759 9917 

37.378 9648 

41 .446 2632 

19 

20 

33.065 9541 

36.785 5912 

40.995 4923 

45.761 9643 

20 

21 

35.719 2518 

39.992 7267 

44.865 1768 

50.422 9214 

21 


38.505 2144 

43.392 2903 

49.005 7392 

55.456 7552 

22 

23 

41.430 4751 

46.995 8277 

53.436 1409 

60.893 2956 

23 

24 

44.501 9989 

50.815 5774 

58.176 6708 

66.764 7592 

24 

25 

47.727 0988 

54.864 5120 

63.249 0377 

73.105 9400 

25 

26 

51.113 4538 

59.156 3827 

68.676 4704 

79.954 4152 

26 

27 

54.669 1264 

63.705 7657 

74.483 8233 

87.350 7684 

27 

28 

58.402 5828 

68.528 1116 

80.697 6909 

95.338 8298 

28 

29 

62.322 7119 

73.639 7983 

87.346 5293 

103.965 9362 

29 

30 

66.438 8475 

79.058 1862 

94.460 7863 

113.283 2111 

30 

31 

70.760 7899 

84.801 6774 

102.073 0414 

123.345 8680 

31 

32 

75.298 8294 

90.889 7780 

110.218 1543 

134.213 5374 

22 

33 

80.063 7708 

97.343 1647 

118.933 4251 

145.950 6204 

33 

34 

85.066 9594 

104.183 7546 

128.258 7648 

158.626 6701 

34 

35 

90.320 3074 

111.434 7799 

138.236 8784 

172.316 8037 

35 

36 

95.836 3227 

119.120 8667 

148.913 4598 

187.102 1480 

36 

37 

101.628 1389 

127.268 1187 

160.337 4020 

203.070 3198 

37 

38 

107.709 5458 

135.904 2058 

172.561 0202 

220.315 9454 

38 

39 

114.095 0231 

145.058 4581 

185.640 2916 

238.941 2210 

39 

40 

120.799 7742 

154.761 9656 

199.635 1120 

259.056 5187 

40 

41 

127.839 7630 

165.047 6836 

214.609 5698 

280.781 0402 

41 

42 

135.231 7511 

175.950 5446 

230.632 2397 

304.243 5234 

42 

43 

142.993 3387 

187.507 5772 

247.776 4965 

329.583 0053 

43 

44 

151.143 0056 

199.758 0319 

266.120 8512 

356.949 6457 

44 

45 

159.700 1559 

212.743 5138 

285.749 3108 

386.505 6174 

45 

46 

168.685 1637 

226.508 1246 

306.751 7626 

418.426 0668 

46 

47 

178.119 4218 

241.098 6121 

329.224 3860 

452.900 1521 

47 

48 

188.025 3929 

256.564 5288 

353.270 0930 

490.132 1643 

48 

49 

198.426 6626 

272.958 4006 

378.998 9995 

530.342 7374 

49 

50 

209.347 9957 

290.335 9046 

406.528 9295 

573.770 1564 

50 

60 

353.583 7179 

533.128 1809 

813.520 3834 

1253.213 2958 

60 

70 

588.528 5107 

967.932 1696 

1614.134 1742 

2720.080 0738 

70 

80 

971 .228 8213 

1746.599 8914 

3189.062 6797 

5886.935 4283 

80 

90 

1594.607 3010 

3141.075 1872 

6287.185 4268 

12723.938 6160 

90 

100 

2610.025 1569 

5638.368 0586 

12381.661 7938 

27484.515 7043 

100 



















108 


TABLES 


TABLE VI—ANNUITY WHICH 1 WILL BUY 



n\ n\ 


n 

y 2 % 

1% 

1 V*% 

i y 2 % 

n 

1 

1.005 0000 

1.010 0000 

1.012 5000 

1.015 0000 

1 

2 

0.503 7531 

0.507 5124 

0.509 3944 

0.511 2779 

2 

3 

0.336 6722 

0.340 0221 

0.341 7012 

0.343 3830 

3 

4 

0.253 1328 

0.256 2811 

0.257 8610 

0.259 4448 

4 

5 

0.203 0100 

0.206 0398 

0.207 5621 

0.209 0893 

5 

6 

0.169 5955 

0.172 5484 

0.174 0338 

0.175 5252 

6 

7 

0.145 7285 

0.148 6283 

0.150 0887 

0.151 5562 

7 

8 

0.127 8289 

0.130 6903 

0.132 1331 

0.133 5840 

8 

9 

0.113 9074 

0.116 7404 

0.118 1706 

0.119 6098 

9 

10 

0.102 7706 

0.105 5821 

0.107 0031 

0.108 4342 

10 

11 

0.093 6590 

0.096 4541 

0.097 8684 

0.099 2938 

11 

12 

0.086 0664 

0.088 8488 

0.090 2583 

0.091 6800 

12 

13 

0.079 6422 

0.082 4148 

0.083 8210 

0.085 2404 

13 

14 

0.074 1361 

0.076 9012 

0.078 3052 

0.079 7233 

14 

15 

0.069 3644 

0.072 1238 

0.073 5265 

0.074 9444 

15 

16 

0.065 1894 

0.067 9446 

0.069 3467 

0.070 7651 

16 

17 

0.061 5058 

0.064 2581 

0.065 6602 

0.067 0796 

17 

18 

0.058 2317 

0.060 9820 

0.062 3848 

0.063 8058 

18 

19 

0.055 3025 

0.058 0518 

0.059 4555 

0.060 8785 

19 

20 

0.052 6664 

0.055 4153 

0.056 8204 

0.058 2457 

20 

21 

0.050 2816 

0.053 0308 

0.054 4375 

0.055 8655 

21 

22 

0.048 1138 

0.050 8637 

0.052 2724 

0.053 7033 

22 

23 

0.046 1346 

0.048 8858 

0.050 2967 

0.051 7308 

23 

24 

0.044 3206 

0.047 0735 

0.048 4866 

0.049 9241 

24 

23 

0.042 6519 

0.045 4068 

0.046 8225 

0.048 2634 

25 

26 

0.041 1116 

0.043 8689 

0.045 2873 

0.046 7320 

26 

27 

0.039 6856 

0.042 4455 

0.043 8668 

0.045 3153 

27 

28 

0.038 3617 

0.041 1244 

0.042 5486 

0.044 0011 

28 

29 

0.037 1291 

0.039 8950 

0.041 3223 

0.042 7788 

29 

30 

0.035 9789 

0.038 7481 

0.040 1785 

0.041 6392 

30 

31 

0.034 9030 

0.037 6757 

0.039 1094 

0.040 5743 

31 

32 

0.033 8945 

0.036 6709 

0.038 1079 

0.039 5771 

32 

33 

0.032 9473 

0.035 7274 

0.037 1679 

0.038 6414 

33 

34 

0.032 0559 

0.034 8400 

0.036 2839 

0.037 7619 

34 

35 

0.031 2155 

0.034 0037 

0.035 4511 

0.036 9336 

35 

36 

0.030 4219 

0.033 2143 

0.034 6653 

0.036 1524 

36 

37 

0.029 6714 

0.032 4680 

0.033 9227 

0.035 4144 

37 

38 

0.028 9604 

0.031 7615 

0.033 2198 

0.034 7161 

38 

39 

0.028 2861 

0.031 0916 

0.032 5536 

0.034 0546 

39 

40 

0.027 6455 

0.030 4556 

0.031 9214 

0.033 4271 

40 

41 

0.027 0363 

0.029 8510 

0.031 3206 

0.032 8311 

41 

42 

0.026 4562 

0.029 2756 

0.030 7491 

0.032 2643 

42 

43 

0.025 9032 

0.028 7274 

0.030 2047 

0.031 7246 

43 

44 

0.025 3754 

0.028 2044 

0.029 6856 

0.031 2104 

44 

45 

0.024 8712 

0.027 7050 

0.029 1901 

0.030 7198 

45 

46 

0.024 3889 

0.027 2278 

0.028 7168 

0.030 2512 

46 

47 

0.023 9273 

0.026 7711 

0.028 2641 

0.029 8034 

47 

48 

0.023 4850 

0.026 3338 

0.027 8307 

0.029 3750 

48 

49 

0.023 0609 

0.025 9147 

0.027 4156 

0.028 9648 

49 

50 

0.022 6538 

0.025 5127 

0.027 0176 

0.028 5717 

50 

60 

0.019 3328 

0.022 2444 

0.023 7899 

0.025 3934 

60 

70 

0.016 9666 

0.019 9328 

0.021 5194 

0.023 1724 

70 

80 

0.015 1970 

0.018 2188 

0.019 8465 

0.021 5483 

80 

90 

0.013 8253 

0.016 9031 

0.018 5715 

0.020 3211 

90 

100 

0.0127319 

0.015 8657 

0.017 5743 

0.019 3706 

100 













ANNUITY WHICH 1 WILL BUY 


109 


TABLE VI—ANNUITY WHICH 1 WILL BUY —Continued 



«! n\ 


n 

l 3 /4% 

2% 

2 Vz% 

3% 

n 

1 

1.017 5000 

1.020 0000 

1.025 0000 

1.030 0000 

1 

2 

0.513 1630 

0.515 0495 

0.518 8272 

0.522 6108 

2 

3 

0.345 0675 

0.346 7547 

0.350 1372 

0.353 5304 

3 

4 

0.261 0324 

0.262 6238 

0.265 8179 

0.269 0270 

4 

5 

0.210 6214 

0.212 1584 

0.215 2469 

0.218 3546 

5 

6 

0.177 0226 

0.178 5258 

0.181 5500 

0.184 5975 

6 

7 

0.153 0306 

0.154 5120 

0.157 4954 

0.160 5064 

7 

8 

0.135 0429 

0.136 5098 

0.139 4674 

0.142 4564 

8 

9 

0.121 0581 

0.122 5154 

0.125 4569 

0.128 4339 

9 

10 

0.109 8754 

0.111 3265 

0.114 2588 

0.117 2305 

10 

11 

0.100 7304 

0.102 1779 

0.105 1060 

0.108 0774 

11 

12 

0.093 1138 

0.094 5596 

0.097 4871 

0.100 4621 

12 

13 

0.086 6728 

0.088 1184 

0.091 0483 

0.094 0295 

13 

14 

0.081 1556 

0.082 6020 

0.085 5365 

0.088 5263 

14 

15 

0.076 3774 

0.077 8255 

0.080 7665 

0.083 7666 

15 

10 

0.072 1996 

0.073 6501 

0.076 5990 

0.079 6108 

16 

17 

0.068 5162 

0.069 9698 

0.072 9278 

0.075 9525 

17 

18 

0.065 2449 

0.066 7021 

0.069 6701 

0.072 7087 

18 

10 

0.062 3206 

0.063 7818 

0.066 7606 

0.069 8139 

19 

20 

0.059 6912 

0.061 1567 

0.064 1471 

0.067 2157 

20 

21 

0.057 3146 

0.058 7848 

0.061 7873 

0.064 8718 

21 

22 

0.055 1564 

0.056 6314 

0.059 6466 

0.062 7474 

22 

23 

0.053 1880 

0.054 6681 

0.057 6964 

0.060 8139 

23 

24 

0.051 3856 

0.052 8711 

0.055 9128 

0.059 0474 

24 

25 

0.049 7295 

0.051 2204 

0.054 2759 

0.057 4279 

25 

26 

0.048 2021 

0.049 6992 

0.052 7688 

0.055 9383 

26 

27 

0.046 7908 

0.048 2931 

0.051 3769 

0.054 5642 

27 

28 

0.045 4815 

0.046 9897 

0.050 0879 

0.053 2932 

28 

29 

0.044 2642 

0.045 7784 

0.048 8913 

0.052 1147 

29 

30 

0.043 1298 

0.044 6499 

0.047 7776 

0.051 0193 

30 

31 

0.042 0700 

0.043 5964 

0.046 7390 

0.049 9989 

31 

32 

0.041 0781 

0.042 6106 

0.045 7683 

0.049 0466 

32 

33 

0.040 1478 

0.041 6865 

0.044 8594 

0.048 1561 

33 

34 

0.039 2736 

0.040 8187 

0.044 0068 

0.047 3220 

34 

35 

0.038 4508 

0.040 0022 

0.043 2056 

0.046 5393 

35 

36 

0.037 6751 

0.039 2328 

0.042 4516 

0.045 8038 

36 

37 

0.036 9426 

0.038 5068 

0.041 7409 

0.045 1116 

37 

38 

0.036 2499 

0.037 8206 

0.041 0701 

0.044 4593 

38 

39 

0.035 5940 

0.037 1711 

0.040 4362 

0.043 8438 

39 

40 

0.034 9721 

0.036 5558 

0.039 8362 

0.043 2624 

40 

41 

0.034 3817 

0.035 9719 

0.039 2679 

0.042 7124 

41 

42 

0.033 8206 

0.035 4173 

0.038 7288 

0.042 1917 

42 

43 

0.033 2867 

0.034 8899 

0.038 2169 

0.041 6981 

43 

44 

0.032 7781 

0.034 3879 

0.037 7304 

0.041 2298 

44 

45 

0.032 2932 

0.033 9096 

0.037 2675 

0.040 7852 

45 

46 

0.031 8304 

0.033 4534 

0.036 8268 

0.040 3625 

46 

47 

0 031 3884 

0.033 0179 

0.036 4067 

0.039 9605 

47 

48 

0 030 9657 

0.032 6018 

0.036 0060 

0.039 5778 

48 

49 

0.030 5612 

0.032 2040 

0.035 6235 

0.039 2131 

49 

50 

0.030 1739 

0.031 8232 

0.035 2581 

0.038 8655 

50 

60 

0 027 0534 

0.028 7680 

0.032 3534 

0.036 1330 

60 

70 

0 024 8893 

0.026 6676 

0.030 3971 

0.034 3366 

i 0 

80 

0 023 3209 

0.025 1607 

0.029 0260 

0.033 1118 

80 

90 

0 022 1476 

0.024 0460 

0.028 0381 

0.032 2556 

90 

100 

0.021 2488 

0.023 2027 

0.027 3119 

0.031 6467 

^ J 

100 


<•♦7 























110 


TABLES 


TABLE VI—ANNUITY WHICH 1 WILL BUY —Continued 


n 

3 V 2 % 

4% 

4%% 

4%% 

n 

1 

1.035 0000 

1.040 0000 

1.045 0000 

1.047 5000 

1 

2 

0.526 4005 

0.530 1961 

0.533 9976 

0.535 9005 

2 

3 

0.356 9342 

0.360 3485 

0.363 7734 

0.365 4897 

3 

4 

0.272 2511 

0.275 4900 

0.278 7436 

0.280 3759 

4 

5 

0.221 4814 

0.224 6271 

0.227 7916 

0.229 3809 

5 

6 

0.187 6682 

0.190 7619 

0.193 8784 

0.195 4451 

6 

7 

0.163 5445 

0.166 6096 

0.169 7015 

0.171 2574 

7 

8 

0.145 4766 

0.148 5278 

0.151 6096 

0.153 1620 

8 

9 

0.131 4460 

0.134 4930 

0.137 5745 

0.139 1280 

9 

10 

0.120 2414 

0.123 2909 

0.126 3788 

0.127 9370 

10 

11 

0.111 0920 

0.114 1490 

0.117 2482 

0.118 8134 

11 

12 

0.103 4840 

0.106 5522 

0.109 6662 

0.111 2402 

12 

13 

0.097 0616 

0.100 1437 

0.103 2754 

0.104 8595 

13 

14 

0.091 5707 

0.094 6690 

0.097 8203 

0.099 4156 

14 

15 

0.086 8251 

0.089 9411 

0.093 1138 

0.094 7211 

15 

16 

0.082 6848 

0.085 8200 

0.089 0154 

0.090 6353 

16 

17 

0.079 0431 

0.082 1985 

0.085 4176 

0.087 0506 

17 

18 

0.075 8168 

0.078 9933 

0.082 2369 

0.083 8834 

18 

19 

0.072 9403 

0.076 1386 

0.079 4073 

0.081 0677 

19 

20 

0.070 3611 

0.073 5818 

0.076 8761 

0.078 5505 

20 

21 

0.068 0366 

0.071 2801 

0.074 6006 

0.076 2891 

21 

22 

0.065 9321 

0.069 1988 

0.072 5456 

0.074 2485 

22 

23 

0.064 0188 

0.067 3091 

0.070 6825 

0.072 3997 

23 

24 

0.062 2728 

0.065 5868 

0.068 9870 

0.070 7187 

24 

25 

0.060 6740 

0.064 0120 

0.067 4390 

0.069 1851 

25 

26 

0.059 2054 

0.062 5674 

0.066 0214 

0.067 7819 

26 

27 

0.057 8524 

0.061 2385 

0.064 7195 

0.066 4944 

27 

28 

0.056 6026 

0.060 0130 

0.063 5208 

0.065 3102 

28 

29 

0.055 4454 

0.058 8799 

0.062 4146 

0.064 2183 

29 

30 

0.054 3713 

0.057 8301 

0.061 3915 

0.063 2095 

30 

31 

0.053 3724 

0.056 8554 

0.060 4434 

0.062 2755 

31 

32 

0.052 4415 

0.055 9486 

0.059 5632 

0.061 4093 

32 

33 

0.051 5724 

0.055 1036 

0.058 7445 

0.060 6046 

33 

34 

0.050 7597 

0.054 3148 

0.057 9819 

0.059 8557 

34 

35 

0.049 9984 

0.053 5773 

0.057 2704 

0.059 1579 

35 

36 

0.049 2842 

0.052 8869 

0.056 6058 

0.058 5068 

36 

37 

0.048 6132 

0.052 2396 

0.055 9840 

0.057 8984 

37 

38 

0.047 9821 

0.051 6319 

0.055 4017 

0.057 3293 

38 

39 

0.047 3878 

0.051 0608 

0.054 8557 

0.056 7964 

39 

40 

0.046 8273 

0.050 5235 

0.054 3432 

0.056 2968 

40 

41 

0.046 2982 

0.050 0174 

0.053 8616 

0.055 8279 

41 

42 

0.045 7983 

0.049 5402 

0.053 4087 

0.055 3876 

42 

43 

0.045 3254 

0.049 0899 

0.052 9824 

0.054 9736 

43 

44 

0.044 8777 

0.048 6645 

0.052 5807 

0.054 5842 

44 

45 

0.044 4534 

0.048 2625 

0.052 2020 

0.054 2175 

45 

46 

0.044 0511 

0.047 8820 

0.051 8447 

0.053 8720 

46 

47 

0.043 6692 

0.047 5219 

0.051 5073 

0.053 5463 

47 

48 

0.043 3065 

0.047 1806 

0.051 1886 

0.053 2390 

48 

49 

0.042 9617 

0.046 8571 

0.050 8872 

0.052 9489 

49 

50 

0.042 6337 

0.046 5502 

0.050 6022 

0.052 6749 

50 

60 

0.040 0886 

0.044 2018 

0.048 4543 

0.050 6271 

60 

70 

0.038 4610 

0.042 7451 

0.047 1651 

0.049 4192 

70 

80 

0.037 3849 

0.041 8141 

0.046 3707 

0.048 6888 

80 

90 

0.036 6578 

0.041 2078 

0.045 8732 

0.048 2405 

90 

100 

0.036 1593 

0.040 8080 

0.045 5584 

0.047 9629 

100 


















ANNUITY WHICH 1 WILL BUY 


111 


TABLE YI—ANNUITY WHICH 1 WILL BUY —Continued 


n 

5% 

6% 

7% 

8% 

n 

1 

1.050 0000 

1.060 0000 

1.070 0000 

1.080 0000 

I 

2 

0.537 8049 

0.545 4369 

0.553 0918 

0.560 7692 

2 

3 

0.367 2086 

0.374 1098 

0.381 0517 

0.388 0335 

3 

4 

0.282 0118 

0.288 5915 

0.295 2281 

0.301 9208 

4 

5 

0.230 9748 

0.237 3964 

0.243 8907 

0.250 4564 

5 

6 

0.197 0175 

0.203 3626 

0.209 7958 

0.216 3154 

6 

7 

0.172 8198 

0.179 1350 

0.185 5532 

0.192 0724 

7 

8 

0.154 7218 

0.161 0359 

0.167 4678 

0.174 0148 

8 

9 

0.140 6901 

0.147 0222 

0.153 4865 

0.160 0797 

9 

10 

0.129 5046 

0.135 8680 

0.142 3775 

0.149 0295 

10 

11 

0.120 3889 

0.126 7929 

0.133 3569 

0.140 0763 

11 

12 

0.112 8254 

0.119 2770 

0.125 9020 

0.132 6950 

12 

13 

0.106 4558 

0.112 9601 

0.119 6508 

0.126 5218 

13 

14 

0.101 0240 

0.107 5849 

0.114 3449 

0.121 2968 

14 

15 

0.096 3423 

0.102 9628 

0.109 7946 

0.116 8295 

15 

16 

0.092 2699 

0.098 9521 

0.105 8576 

0.112 9769 

16 

17 

0.088 6991 

0.095 4448 

0.102 4252 

0.109 6294 

17 

18 

0.085 5462 

0.092 3565 

0.099 4126 

0.106 7021 

18 

19 

0.082 7450 

0.089 6209 

0.096 7530 

0.104 1276 

19 

20 

0.080 2426 

0.087 1846 

0.094 3929 

0.101 8522 

20 

21 

0.077 9961 

0.085 0046 

0.092 2890 

0.099 8322 

21 

22 

0.075 9705 

0.083 0456 

0.090 4058 

0.098 0321 

22 

23 

0.074 1368 

0.081 2785 

0.088 7139 

0.096 4222 

23 

24 

0.072 4709 

0.079 6790 

0.087 1890 

0.094 9780 

24 

25 

0.070 9525 

0.078 2267 

0.085 8105 

0.093 6788 

25 

26 

0.069 5643 

0.076 9044 

0.084 5610 

0.092 5071 

26 

27 

0.068 2919 

0.075 6972 

0.083 4257 

0.091 4481 

27 

28 

0.067 1225 

0.074 5926 

0.082 3919 

0.090 4889 

28 

29 

0.066 0455 

0.073 5796 

0.081 4486 

0.089 6185 

29 

30 

0.065 0514 

0.072 6489 

0.080 5864 

0.088 8274 

30 

31 

0.064 1321 

0.071 7922 

0.079 7969 

0.088 1073 

31 

32 

0.063 2804 

0.071 0023 

0.079 0729 

0.087 4508 

32 

33 

0.062 4900 

0.070 2729 

0.078 4081 

0.086 8516 

33 

34 

0.061 7554 

0.069 5984 

0.077 7967 

0.086 3041 

34 

35 

0.061 0717 

0.068 9739 

0.077 2340 

0.085 8033 

35 

36 

0.060 4345 

0.068 3948 

0.076 7153 

0.085 3447 

36 

37 

0.059 8398 

0.067 8574 

0.076 2368 

0.084 9244 

37 

38 

0.059 2842 

0.067 3581 

0.075 7950 

0.084 5389 

38 

39 

0.058 7646 

0.066 8938 

0.075 3868 

0.084 1851 

39 

40 

0.058 2782 

0.066 4615 

0.075 0091 

0.083 8602 

40 

41 

0.057 8223 

0.066 0589 

0.074 6596 

0.083 5615 

41 

42 

0.057 3947 

0.065 6834 

0.074 3359 

0.083 2868 

42 

43 

0.056 9933 

0.065 3331 

0.074 0359 

0.083 0341 

43 

44 

0.056 6162 

0.065 0061 

0.073 7577 

0.082 8015 

44 

45 

0.056 2617 

0.064 7005 

0.073 4996 

0.082 5873 

45 

46 

0.055 9282 

0.064 4148 

0.073 2600 

0.082 3899 

46 

47 

0.055 6142 

0.064 1477 

0.073 0374 

0.082 2080 

47 

48 

0.055 3184 

0.063 8977 

0.072 8307 

0.082 0403 

48 

49 

0.055 0396 

0.063 6636 

0.072 6385 

0.081 8856 

49 

50 

0.054 7767 

0.063 4443 

0.072 4598 

0.081 7429 

50 

60 

0.052 8282 

0.061 8757 

0.071 2292 

0 .080 7980 

60 

70 

0.051 6992 

0.061 0331 

0.070 6195 

0.080 3676 

70 

80 

0.051 0296 

0.060 5725 

0.070 3136 

0.080 1699 

80 

90 

0.050 6271 

0.060 3184 

0.070 1590 

0.080 0786 

90 

100 

0.050 3831 

0.060 1774 

0.070 0808 

0.080 0364 

IlOO 






















112 


TABLES 


TABLE VII—AMOUNT OF 1 FOR PARTS OF A YEAR 

l_ 

s=(l+i)J> 


p 

%% 

1% 

l‘/4% 

1 Vi% 

P 

2 

1.002 4969 

1.004 9876 

1.006 2306 

1.007 4721 

2 

4 

1.001 2477 

1.002 4907 

1.003 1105 

1.003 7291 

4 

12 

1.000 4157 

1.000 8295 

1.001 0357 

1.001 2415 

12 

P 

1% % 

2% 

2 x h% 

3% 

P 

2 

1.008 7121 

1.009 9505 

1.012 4228 

1.014 8892 

2 

4 

1.004 3466 

1.004 9629 

1.006 1922 

1.007 4171 

4 

12 

1.001 4468 

1.001 6516 

1.002 0598 

1.002 4663 

12 

P 

3V4% 

4% 

4 y 2 % 

4 3/ 4 % 

P 

2 

1.017 3495 

1.019 8039 

1.022 2524 

1.023 4745 

2 

4 

1.008 6374 

1.009 8534 

1.011 0650 

1 011 6692 

4 

12 

1.002 8709 

1.003 2737 

1.003 6748 

1.003 8747 

12 

P 

5% 

a% 

7% 

8% 

P 

2 

1.024 6951 

1.029 5630 

1.034 4080 

1.039 2305 

2 

4 

1.012 2722 

1.014 6738 

1.017 0585 

1.019 4265 

4 

12 

1.004 0741 

1.004 8676 

1.005 6541 

1.006 4340 

12 


l_ 

TABLE VIII—VALUES OF j {p) = />[(! + i)P - l] 


P 

%% 

1% 

1 Vi % 

4 1/2% 

P 

2 

0.004 9938 

0.009 9751 

0.012 4612 

0.014 9442 

2 

4 

0.004 9907 

0.009 9627 

0.012 4418 

0.014 9164 

4 

12 

0.004 9886 

0.009 9545 

0.012 4290 

0.014 8978 

12 

P 

l 3 /4% 

2% 

21/2% 

3% 

P 

2 

0.017 4241 

0.019 9010 

0.024 8457 

0.029 7783 

2 

4 

0.017 3863 

0.019 8517 

0.024 7690 

0.029 6683 

4 

12 

0.017 3612 

0.019 8190 

0.024 7180 

0.029 5952 

12 

P 

sy 2 % 

4% 

4i/ 2 % 

4 3/ 4 % 

P 

2 

0.034 6990 

0.039 6078 

0.044 5048 

0.046 9489 

2 

4 

0.034 5498 

0.039 4136 

0.044 2600 

0.046 6766 

4 

12 

0.034 4508 

0.039 2849 

0.044 0977 

0.046 4962 

12 

P 

5% 

6% 

7% 

8% 

P 

2 

0.049 3902 

0.059 1260 

0.068 8161 

0.078 4610 

2 

4 

0.049 0889 

0.058 6954 

0.068 2341 

0.077 7062 

4 

12 

0.048 8895 

0.058 4106 

0.067 8497 

0.077 2084 

12 


TABLE IX—VALUES OF J— 

hp) 


P 

1 / 2 % 

4% 

1 Vi% 

iy 2 % 

P 

2 

1.001 2415 

1.002 4938 

1.003 1153 

1.003 7360 

2 

4 

1.001 8635 

1.003 7422 

1.004 6754 

1.005 6076 

4 

12 

1.002 2852 

1.004 5751 

1.005 7163 

1.006 8565 

12 

P 

l 3 /4% 

2% 

2y z % 

3% 

P 

2 

1.004 3560 

1.004 9753 

1.006 2114 

1.007 4446 

2 

4 

1.006 5388 

1.007 4691 

1.009 3268 

1.011 1807 

4 

12 

1.007 9957 

1.009 1339 

1.011 4073 

1.013 6766 

12 

P 

31 / 2 % 

4% 

4%% 

4 3/4% 

P 

2 

1.008 6748 

1.009 9020 

1.011 1262 

1.011 7383 

2 

4 

1.013 0309 

1.014 8774 ' 

1.016 7203 

1.017 6405 

4 

12 

1.015 9420 

1.018 2035 

1.020 4611 

1.021 5889 

12 

P 

5% 

6% 

7% 

8% 

P 

2 

1.012 3475 

1.014 7815 

1.017 2040 

1.019 6148 

2 

4 

1.018 5594 

1 022 2269 

1.025 8800 

1.029 5189 

4 

12 

1.022 7148 

_ ■* 

1.027 2107 

1.031 6914 

1.036 1567 

12 












































































































































AMERICAN EXPERIENCE TABLE OF MORTALITY 113 


TABLE X—AMERICAN EXPERIENCE TABLE OF MORTALITY 


Age 

X 

Num¬ 

ber 

living 

lx 

Num¬ 

ber 

of 

deaths 

d x 

Yearly 
proba¬ 
bility of 
dying 

Qx 

Yearly 
proba¬ 
bility of 
living 

Vx 

1 

Age 

X 

Num¬ 

ber 

living 

lx 

Num¬ 

ber 

of 

deaths 

d x 

Yearly 
proba¬ 
bility of 
dying 

Qx 

Yearly 
proba¬ 
bility of 
living 

Vx 

10 

100,000 

749 

0.007 490 

0.992 510 

53 

66,797 

1,091 

0.016 333 

0.983 667 

11 

99,251 

746 

0.007 516 

0.992 484 

54 

65,706 

1,143 

0.017 396 

0.982 604 

12 

98,505 

743 

0.007 543 

0.992 457 

55 

64,563 

1,199 

0.018 571 

0.981 429 

13 

97,762 

740 

0.007 569 

0.992 431 

56 

63,364 

1,260 

0.019 885 

0.980 115 

14 

97,022 

737 

0.007 596 

0.992 404 

57 

62,104 

1,325 

0.021 335 

0.978 665 

15 

96,285 

735 

0.007 634 

0.992 366 

58 

60,779 

1,394 

0.022 936 

0.977 064 

16 

95,550 

732 

0.007 661 

0.992 339 

59 

59,385 

1,468 

0.024 720 

0.975 280 

17 

94,818 

729 

0.007 688 

0.992 312 

60 

57,917 

1,546 

0.026 693 

0.973 307 

18 

94,089 

727 

0.007 727 

0.992 273 

61 

56,371 

1,628 

0.028 880 

0.971 120 

19 

93,362 

725 

0.007 765 

0.992 235 

62 

54,743 

1,713 

0.031 292 

0.968 708 

20 

92,637 

723 

0.007 805 

0.992 195 

63 

53,030 

1,800 

0.033 943 

0.966 057 

21 

91,914 

722 

0.007 855 

0.992 145 

64 

51,230 

1,889 

0.036 873 

0.963 127 

22 

91,192 

721 

0.007 906 

0.992 094 

65 

49,341 

1,980 

0.040 129 

0.959 871 

23 

90,471 

720 

0.007 958 

0.992 042 

66 

47,361 

2,070 

0.043 707 

0.956 293 

24 

89,751 

719 

0.008 011 

0.991 989 

67 

45,291 

2,158 

0.047 647 

0.952 353 

25 

89,032 

718 

0.008 065 

0.991 935 

68 

43,133 

2,243 

0.052 002 

0.947 998 

26 

88,314 

718 

0.008 130 

0.991 870 

69 

40,890 

2,321 

0.056 762 

0.943 238 

27 

87,596 

718 

0.008 197 

0.991 803 

70 

38,569 

2,391 

0.061 993 

0.938 007 

28 

86,878 

718 

0.008 264 

0.991 736 

71 

36,178 

2,448 

0.067 665 

0.932 335 

29 

86,160 

719 

0.008 345 

0.991 655 

72 

33,730 

2,487 

0.073 733 

0.926 267 

30 

85,441 

720 

0.008 427 

0.991 573 

73 

31,243 

2,505 

0.080 178 

0.919 822 

31 

84,721 

721 

0.008 510 

0.991 490 

74 

28,738 

2,501 

0.087 028 

0.912 972 

32 

84,000 

723 

0.008 607 

0.991 393 

75 

26,237 

2,476 

0.094 371 

0.905 629 

33 

83,277 

726 

0.008 718 

0.991 282 

76 

23,761 

2,431 

0.102 311 

0.897 689 

34 

82,551 

729 

0.008 831 

0.991 169 

77 

21,330 

2,369 

0.111 064 

0.888 936 

35 

81,822 

732 

0.008 946 

0.991 054 

78 

18,961 

2,291 

0.120 827 

0.879 173 

36 

81,090 

737 

0.009 089 

0.990 911 

79 

16,670 

2,196 

0.131 734 

0.868 266 

37 

80,353 

742 

0.009 234 

0.990 766 

80 

14,474 

2,091 

0.144 466 

0.855 534 

38 

79,611 

749 

0.009 408 

0.990 592 

81 

12,383 

1,964 

0.158 605 

0.841 395 

39 

78,862 

756 

0.009 586 

0.990 414 

82 

10,419 

1,816 

0.174 297 

0.825 703 

40 

78,106 

765 

0.009 794 

0.990 206 

83 

8,603 

1,648 

0.191 561 

0.808 439 

41 

77,341 

774 

0.010 008 

0.989 992 

84 

6,955 

1,470 

0.211 359 

0.788 641 

42 

76,567 

785 

0.010 252 

0.989 748 

85 

5,485 

1,292 

0.235 552 

0.764 448 

43 

75,782 

797 

0.010 517 

0.989 483 

86 

4,193 

1,114 

0.265 681 

0.734 319 

44 

74,985 

812 

0.010 829 

0.989 171 

87 

3,079 

933 

0.303 020 

0.696 980 

45 

74,173 

828 

0.011 163 

0.988 837 

88 

2,146 

744 

0.346 692 

0.653 308 

46 

73,345 

848 

0.011 562 

0.988 438 

89 

1,402 

555 

0.395 863 

0.604 137 

47 

72,497 

870 

0.012 000 

0.988 000 

90 

847 

385 

0.454 545 

0.545 455 

48 

71,627 

896 

0.012 509 

0.987 491 

91 

462 

246 

0.532 466 

0.467 534 

49 

70,731 

927 

0.013 106 

0.986 894 

92 

216 

137 

0.634 259 

0.365 741 

50 

69,804 

962 

0.013 781 

0.986 219 

93 

79 

58 

0.734 177 

0.265 823 

51 

68,842 

1,001 

0.014 541 

0.985 459 

94 

21 

18 

0.857 143 

0.142 857 

52 

67,841 

1,044 

0.015 389 

0.984 611 

95 

3 

3 

1.000 000 

0.000 000 


































114 


TABLES 


TABLE XI—COMMUTATION COLUMNS 


American Experience, 3 y 2 % 


0) 

be 

< 

X 

D x 

N x 

M x 

h Age 

D x 

N x 

M x 

10 

70 891 .9 

1 575 535 .3 

17 612.91 

53 

10 787 .4 

145 915 .7 

5 853 .095 

11 

67 981 .5 

1 504 643 .4 

17 099 .89 

54 

10 252 .4 

135 128 .2 

5 682 .861 

12 

65 189 .0 

1 436 661 .9 

16 606 .20 

55 

9 733 .40 

124 875 .8 

5 510 .544 

13 

62 509 .4 

1 371 472 .9 

16 131 .12 

56 

9 229 .60 

115 142 .4 

5 335 .898 

14 

59 938 .4 

1 308 963 .5 

15 673 .96 

57 

8 740.17 

105 912.8 

5 158 .573 

15 

54 471 .6 

1 249 025 .0 

15 234.05 

58 

8 264 .44 

97 172 .64 

4 978 .405 

16 

55 104 .2 

1 191 553 .4 

14 810.17 

59 

7 801 .83 

88 908 .20 

4 795 .266 

17 

52 832 .9 

1 136 449 .2 

14 402 .30 

60 

7 351 .65 

81 106 .38 

4 608 .926 

18 

50 653 .9 

1 083 616.2 

14 009 .83 

61 

6 913 .44 

73 754 .73 

4 419 .322 

19 

48 562.8 

1 032 962 .4 

13 631 .68 

62 

6 486 .75 

66 841 .28 

4 226 .413 

20 

46 556 .2 

984 399 .6 

13 267 .32 

63 

6 071 .27 

60 354 .54 

4 030 .296 

21 

44 630 .8 

937 843 .3 

12 916 .25 

64 

5 666 .85 

54 283 .27 

3 831 .187 

22 

42 782 .8 

893 212.5 

12 577 .53 

65 

5 273 .33 

48 616 .41 

3 629 .300 

23 

41 009 .2 

850 429 .7 

12 250 .71 

66 

4 890 .55 

43 343 .08 

3 424 .843 

24 

39 307 .1 

809 420 .5 

11 935 .38 

67 

4 518.65 

38 452 .53 

3 218 .321 

25 

37 673 .6 

770 113 .4 

11 631 .14 

68 

4 157 .82 

33 933 .88 

3 010 .299 

26 

36 106 .1 

732 439 .8 

11 337 .59 

69 

3 808 .32 

29 776 .06 

2 801 .396 

27 

34 601 .5 

696 333 .7 

11 053 .97 

70 

3 470 .67 

25 967 .74 

2 592 .538 

28 

33 157 .4 

661 733 .2 

10 779 .94 

71 

3 145 .43 

22 497 .07 

2 384 .657 

29 

31 771 .3 

628 574 .8 

10 515.18 

72 

2 833 .42 

19 351 .64 

2 179 .018 

30 

30 440 .8 

596 803 .6 

10 259 .02 

73 

2 535 .75 

16 518.22 

1 977 .167 

31 

29 163 .5 

566 362 .9 

10 011 .17 

74 

2 253 .57 

13 982.47 

1 780 .731 

32 

27 937 .5 

537 199.3 

9 771 .375 

75 

1 987 .87 

11 728.90 

1 591 .240 

33 

26 760.5 

509 261 .8 

9 539 .044 

76 

1 739.39 

9 741 .028 

1 409 .988 

34 

25 630.1 

482 501 .3 

9 313 .638 

77 

1 508 .63 

8 001 .633 

1 238 .047 

35 

24 544 .7 

456 871 .2 

9 094 .955 

78 

1 295 .73 

6 492 .999 

1 076 .158 

36 

23 502.5 

432 326 .5 

8 882 .798 

79 

1 100.65 

5 197 .271 

924 .893 7 

37 

22 501 .4 

408 824 .0 

8 676 .415 

80 

923 .338 

4 096 .624 

784 .804 6 

38 

21 539.7 

386 322 .6 

8 475 .658 

81 

763 .234 

3 173 .286 

655 .924 5 

39 

20 615.5 

364 782.9 

8 279.860 

82 

620 .465 

2 410 .052 

538 .965 7 

40 

19 727.4 

344 167 .4 

8 088.915 

83 

494 .995 

1 789.587 

434 .477 6 

41 

18 873 .6 

324 440 .0 

7 902 .231 

84 

386 .641 

1 294 .592 

342 .862 4 

42 

18 052 .9 

305 566 .3 

7 719.738 

85 

294 .610 

907 .951 3 

263 .905 9 

43 

17 263 .6 

287 513 .4 

7 540 .910 

86 

217 .598 

613 .341 7 

196 .856 9 

44 

16 504.4 

270 249.8 

7 365 .489 

87 

154 .383 

395 .743 8 

141 .000 3 

45 

15 773 .6 

253 745 .5 

7 192 .809 

88 

103 .963 

241 .360 9 

95 .801 07 

46 

15 070 .0 

237 971 .9 

7 022 .682 

89 

65 .623 1 

137 .397 8 

60 .976 81 

47 

14 392 .1 

222 901 .9 

6 854 .337 

90 

38 .304 7 

71 .774 70 

35 .877 55 

48 

13 738.5 

208 509 .8 

6 687 .466 

91 

20 .186 9 

33 .470 01 

19 .055 09 

49 

13 107 .9 

194 771 .3 

6 521 .419 

92 

9 .118 89 

13 .283 09 

8 .669 695 

50 

12 498.6 

181 663 .4 

6 355 .436 

93 

3 .222 36 

4 .164 21 

3 .081 545 

51 

11 909 .6 

169 164 .7 

6 189 .012 

94 

0 .827 611 

0 .941 84 

0 .795 762 

52 

11 339 .5 

157 255.2 

6 021 .696 

95 

0 .114 232 

0.114 23 

0 .110 369 



















INDEX 

(Numbers refer to pages.) 


A 

Accumulation of discount, 56 
schedule, 57 

American Experience Mortality 
Table, 113 
Amortization, 33, 38 
of premium on bond, 54 
Amount, 2 
compound, 4 
of annuity, 21 
of unpaid principal, 34 
table for, 104 
table for compound, 92 
Annuity, 16 

amount of, 21, 104 
certain, 10, 75 
contingent, 75 
deferred, 23 
deferred life, 77 
due, 23, 79 
life, 74 

present value of, 16, 19, 100 
table for, 108 
temporary, 78, 80 
that 1 will purchase, 25 
that will amount to, 26 

B 

Binomial theorem, 65 
Bonds, 49 
discount on, 52 
investment rate, 49, 59 
par value of, 49 
premium and discount, 52 


Bonds, purchase price of, 57 
redemption of, 49 
valuation of, 50 
Book value, 40 

C 

Capitalization, 28 
Capitalized cost, 28 
Combinations, 64 
Commutation columns, 77 
table of, 114 
Composite life, 44 
Compound interest, 3 
formulas of, 3 

Computation of annuities, 17, 21 
by logarithms, 17, 18 
of compound interest, 4, 5 
of simple interest, 2 
Conversion period, 4 

D 

Depreciation, 39, 40, 42 
fund, 40 
Discount, 12 
on bond, 52 
rate of, 12 

E 

Effective rate of interest, 7 
Endowment, 74 
insurance, 87 

Equation of payments, 14 
Equivalence, principle of, 13 

115 



116 


INDEX 


F 

Force of interest, 8 
G 

Geometrical representation of in¬ 
terest, 6 

I 

Insurance, 81 
endowment, 87 
term, 81 
whole life, 81 

Interest, 1 
compound, 3 

computation of compound, 4, 6 

computation of simple, 1, 2 

continuous compound, 8 

exact, 2 

force of, 8 

ordinary, 2 

rate of, 1 

simple, 1 

J 

Joint life probabilities, 71 

L 

Life annuity, 74 

computation of, 75 
deferred, 77 
due, 79 

present value of, 75 
temporary, 78 

Limited-payment life policy, 84 

Loading, 82, 89 

M 

Mathematical expectation, 70 

Mining properties, 
valuation of, 45 

Mortality tables, 70 

American experience, 113 


N 

Net annual premium, 84 

for endowment insurance, 87 
for n-payment life policy, 84 
for ordinary life policy, 84 
for term insurance, 86 
Net premium, 87 
Net single premium, 82 

for endowment insurance, 87 
for term insurance, 85 
for whole life insurance, 82 
Nominal rate of interest, 7 
Number of day of year, 91 

P 

Par value of bond, 49 
Permutations, 64 
Perpetuity, 28 
Policy of insurance, 81 
limited payment, 84 
ordinary life, 84 
year, 82 

Premium, life insurance, 81 
annual, 84 
gross, 82, 89 
level, 88 
net, 81, 82 
office, 82 
on bond, 52 
amortization of, 54 
computation of, 52 
Present value, 10 

of annuity, 16, 19, 100 
of deferred annuity, 23 
of deferred life annuity, 77 
of endowment, 87 
of life annuity, 75, 77 
of perpetuity, 28 
Probability, 62 
compound, 67 
joint life, 71 

mutually exclusive events, 66 




INDEX 


117 


Probability, repeated trials, 68 
Probable life, 40 

R 

Rate of discount, 12 
of interest, 1 
effective, 7 
nominal, 7 

realized by investor, 49, 58 
Redemption fund, 45, 46 
Reserve, terminal, 88 
Residual value, 40 

S 

Schedule for amortization, 33 
of premium on bond, 55 
Scrap value, 40 
Series, 

binomial, 65 
logarithmic, 9 


Sinking fund, 35, 37 
Surplus, 90 

T 

Temporary annuity, 78 
present value of, 78 
Term insurance, 81 
premium for, 85 
Terminal reserve, 88 

V 

Valuation of bonds, 50 
of mining properties, 45 
of policy, 88 
Value, depreciable, 40 
of mine, 45 
residual, 46 
scrap, 40 
wearing, 40, 44 

W 

Whole life insurance, 81, 82 






















































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